APT PRINT

APT or Automatically Programmed Tool is a high-level computer programming language used to generate

instructions for numerically controlled machine tools. Douglas T. Ross is considered by many to be the

father of APT. APT is a language and system that makes numerically controlled manufacturing possible.

This

early language was used widely into the 1970s and is still a standard internationally.

APT is used to program CNC machine tools to create complex parts using a cutting tool moving in space. It

is used to calculate a point-to-point path that a tool must follow to generate a desired form. APT is a CAM

system based on a special-purpose language. It was created and refined during the late 1950s and early

1960s to simplify the task of calculating geometry points that a tool must traverse in space to cut the

increasingly complex parts required in the aerospace industry. It was a direct result of the new CNC

technology becoming available at that time, and the daunting task that a machinist or engineer faced

calculating the movements of the CNC for the complex parts for which it was capable. APT was created

before graphical interfaces were available, and so it relies on text to specify the geometry and toolpaths

needed to machine a part.

APT shares many similarities to computer programming languages like Fortran. A general-purpose

computer

language takes source text and converts the statements to instructions that can operate the internals of a

computer. APT converts source statements into programs for numerically- or computer-controlled machine

tools. Typically, this is a text file that contains CNC-vendor dependent commands to generate tool motions

and machine states. Most commonly, this is some form of G-code.

Chapter 1. Language and Syntax

The input to the APT processor is a part program that is a collection of APT statements set forth in an

ordered fashion. APT statements are used for the following purposes:

To define a scalar or geometric quantity.

To describe a cutter path.

To describe auxiliary machine tool functions.

To define machining parameters.

These statements are typed into an editor and saved as a plain-text ASCII file, which is the input to the APT

processor. Columns 1 through 72, inclusively, contain information pertinent to the part programming text,

while columns 73 to 80 are reserved for sequence and/or identification purposes. Blank columns are ignored

by the processor, so blanks can be used to make the text more readable.

1.1. Elements of the APT Language

Each statement is made up of one or more of the following elements (expressed according to set rules of

syntax):

Punctuation

Vocabulary Words

Numbers

Symbols, including Labels

1.1.1. Punctuation

Punctuation for the APT language includes the comma, slash, asterisk, double asterisk, plus, minus, single

dollar, double dollar, equal, period and parentheses [, / * ** + - $ $$ = . ( )]. This punctuation is used to

separate individual words and elements of the source statements used in the APT language and to indicate

arithmetic operations. The APT processor will automatically accept either BCD or EBCDIC characters.

(ASCII for apt360)

The Comma (,)

Used to separate elements of a statement where no other punctuation is appropriate.

MACHIN/ BENDIX, 2, CURCUL

TLLFT, GOLFT/ L1, TO, l2

The Slash (/)

Used either to separate major elements of a statement or as the arithmetic operator, divide, in a

compute statement.

GOLFT/ L1, PAST, C3

A = B / D

The Asterisk (*)

Used as the arithmetic operator multiply.

A = B * C

The Double Asterisk (**)

Used as the arithmetic operator exponentiation. The example below means A=BC.

A = B ** C

The Plus Sign (+)

Used either to indicate arithmetic addition or to specify a signed number. In specifying signed

numbers, plus is assumed if no sign appears.

A = B + C

D = E + F * (+2) * G

P = POINT/ +1, +2, +3

The Minus Sign (-)

Used either to indicate arithmetic subtraction or to specify a signed number.

A = B - C

D = E + F * (-2) * G

P = POINT/ -1, -2, -3

The Single Dollar Sign ($)

Used to indicate the end of a line and to indicate to the processor that the APT statement is continued

on the next line. Any information appearing to the right of the dollar sign is treated as commentary;

that is, the full line is printed out as input translation time, but the information to the right of the

dollar sign is ignored by the rest of the processor.

P1 = PLANE/ PARLEL, PL1, $ clearance

XLARGE, 6.9

A = B * C $ continued on the next line

- D + G

The Double Dollar Sign ($$)

This sign is optional; it's use indicates the end of a part program statement before the end of the 72

column data field. The information to the right of the double dollar sign is to be treated as

commentary. Absence of either a single or double dollar sign causes a part program statement to be

discontinued at the end of the 72-column data field.

CUTTER/ 1, .25 $$ Filleted end mill

The Equal Sign (=)

This sign is used as follows:

$$ To assign a name to a geometric surface or macro.

$$ (This name could be used as a variable later in the source program.)

PT1 = POINT/ X, Y, Z

MAC1 = MACRO/ A, B, C

$$ To assign a value to a scalar variable.

A = B + C

$$ To assign a variable to a macro variable.

CALL/ MAC1, ALPHA1 = 4., ALPHA2 = B

MAC1 = MACRO/ A1 = B, C, D

The Period (Decimal Point) (.)

Used to separate the integral from the fractional part of a number.

4.62

2.

1.23

.005

The Left and Right Parentheses [()]

Used as follows:

$$ To enclose the arguments of an arithmetic function

A = COSF(B)

$$ To enclose subscripts of a variable

A(J) = B(2) + G

$$ To enclose the arithmetic argument of the arithmetic IF statement

IF(A - B * C) 12, 13, 14

$$ To enclose nested definitions

GO/ (L1 = LINE/ X1, Y1, X2, Y2), L3

$$ To enclose any parenthetical expression or related list of objects

A = B * (-3) + c

$$ Right parenthesis -- to separate statement label from the statement

A4) K = 1

1.1.2. Vocabulary Words

The APT processor recognizes a set of vocabulary words that are assigned to the following classes of

activity:

Types of geometric quantities and surfaces; POINT, PLANE, SPHERE, etc.

Computing operators and functions; IF, SINF, COSF, etc.

Postprocessor instructions; STOP, END, SPINDL, etc.

Modifiers, designators and selectors; XLARGE, POSY, LEFT, etc.

Action verbs; GO, GODLTA, etc.

Modes of operation; NOPOST, CLPRNT, PTONLY, etc.

1.1.3. Numbers

All numbers used in a part program are treated by the APT processor as floating-point quantities. Integers

are not treated as a separate class of numbers. The number one (1) may be expressed in the following ways:

1

1.

1.0

.01e2

10.0e-1

1000.e-3

.001e+3

+1

+10.e-1

When writing a number in the exponential format (using an e followed by an exponent), a decimal point

must be given in the number preceding the e. If no decimal point appears, the presence of the alpha character

e will cause the number to be interpreted as a variable symbol.

The maximum number of significant digits allowed is 16, with a range of about 10-75

to 1075

.

Whenever a number is used as an index of a subscript only, its integral part is used; for example, if I = 2.4,

the index or subscript value of I is 2.

1.1.4. Symbols

A symbol is used to define an entity in a part program statement that is to be referenced in later statements.

For example, the statement

CIRC1 = CIRCLE/ 1, 2, 3, 8

defines a circle whose center is at X = 1, Y = 2 and Z = 3, and whose radius is 8. Later part program

statements can now refer to this circle by referencing the symbol CIRC1.

A symbol is made up of from one to six alphanumeric characters, at least one of which must be a letter.

(Punctuation characters cannot be used in a variable symbol.) A symbol, sometimes noted as "variable

symbol", cannot be the same as a vocabulary word. Further, a symbol must be defined before a reference to

that symbol in another part program statement. Because of the dynamic environment of the APT language,

new vocabulary words are frequently required. The use of at least one numeric character in a symbol is

recommended to protect against subsequent changes.

1.1.5. Statement Labels

A statement may have a label (or "ID") so that other part program statements can reference it. A statement

label must be from on the six alphanumeric characters (all of which can be numeric) and must occur to the

extreme left of the statement. A statement label must be terminated by a right parentheses.

IF (I + 2) JOE, 123, 41E

JOE) I = 1

123) J = 3

41E) K = 1

1.1.6. Statement Size Limitation

The maximum number of elements that may be included in a single APT statement is 600. For example, this

statement:

ID6) CIRC1 = CIRCLE/ 1, 2, 3, 8

contains the 13 elements listed in Table 1.1.

Table 1.1. Statement Elements

Count Symbol Type Count Symbol Type

1 ID6 Statement Label 8 , Punctuation

2 ) Punctuation 9 2 Number

3 CIRC1 Symbol 10 , Punctuation

4 = Punctuation 11 3 Number

5 CIRCLE Vocabulary Word 12 , Punctuation

6 / Punctuation 13 8 Number

7 1 Number

Chapter 2. Computing and Subscripted Variables

The ability to perform a variety of computing operations and the concept of subscripted variables have been

included to provide additional capability in the APT language.

2.1. Computing in the APT Language

The computing feature allows a given quantity to be expressed as the result of one or more arithmetic

operations. For example, the statement:

A = 2 + 6 * SQRTF(4)

specifies that the symbol A is to be assigned the value 14 (the result of multiplying 6 times the square root of

4 and adding 2). A computing statement may be made up of some combination of the following:

Scalars

Arithmetic Operators

Arithmetic Expression

Arithmetic Functions

2.1.1. Scalars

Scalars are one-dimensional quantities that can be operated on as is or combined to define a geometric

quantity. Scalars, sometimes noted as "scalar variables", can be multiply-defined, that is, the value of the

scalar can be reassigned by a new computing statement.

2.1.2. Arithmetic Operators

Table 2.1. Operators

Operator Description

+ Addition

- Subtraction

* Multiplication

/ Division

** Exponentiation

2.1.3. Arithmetic Expressions

Scalar symbols, numbers and arithmetic operators may be combined to form an arithmetic expression.

2 * A + B

3. * A + B/ (4. * C)

A + B ** 2 + 3 * C

In the APT language, an arithmetic expression may replace a scalar or a subscript. A scalar symbol or a

variable my be assigned a value by an arithmetic expression.

F = 2 * A + B

C = 3 * A + B / (4. * C)

H = 3.14159

Parentheses may be used in arithmetic expressions to indicate the order of the operations. Within

parentheses, or where parentheses are omitted, the order of operations is as follows:

Table 2.2. Operator Precedence

Operator Precedence

Evaluation of Functions 1 (highest)

** 2

* / 3

+ - 4 (lowest)

Among operations of the same priority, processing is done from left to right.

For example the following expressions are equivalent:

F = A * B + C / D * A ** 2

F = (A * B) + (C / D) * (A ** 2)

An arithmetic expression enclosed in parentheses can be substituted for a scalar value (where a scalar value

is appropriate) in and APT statement except RESERV/.

P1 = POINT/ (A + 2. * B), Y, (-3 + E)

However, an arithmetic expression can never appear to the left of an equal sign, except when used to denote

a subscript value.

Legal Not Legal

A(I + 2) = D * C I + 2 = D * C

D = A(I + K + 1) CALL/ MAC1, B + C = A, D = E / 2, F = G + H

CALL/ MAC2, B = A(I + K * 2)

2.1.4. Arithmetic Functions

The following functions are available in the APT language:

Table 2.3. Arithmetic Functions

Name Function Example Usage

ABSF(x) Absolute Value of x X = ABSF(-22.5)

ATANF(x) Arctangent of x X = ATANF(.577350)

EXPF(x) ex X = EXPF(2)

SINF(x) Sine of x X = SINF(45)

COSF(x) Cosine of x X = COSF(45)

Name Function Example Usage

DOTF(svector, svector) Dot Product of Vectors DP = DOTF(V1, V2)

LNTHF(svector) Vector Magnitude LEN = LNTHF(V)

LOGF(x) logex LOG = LOGF(2)

SQRTF(x) Square Root of x SQR = SQRTF(4)

ANGLF(scircle, spoint)

Angle between the positive X axis

and a line joining the center of

a circle and a point.

A = ANGLF(C1, (POINT/ 2, -

3))

NUMF(SPAT) The number of points in a pattern. N = NUMF(PAT1)

DISTF(splane, splane)

or

DISTF(sline, sline)

Distance between parallel lines

or parallel planes. (Mixing

surfaces types is not allowed.)

D = DISTF(L1, L2)

LOG10F(x) Log10x LG = LOG10F(2)

TANF(x) Tangent of x T = TANF(30.)

ATAN2F(x, y) Arctangent of x/y A = ATAN2F(2, .5)

Note

The arguments to SINF, COSF and TANF are in degrees. ANGLF,

ATANF and ATANF2F return degrees.

The output of each of these functions is single-valued; hence, these functions can be used to define an

arithmetic expression.

R = A * SQRTF(C ** 2 + D ** 2) / E

2.2. Subscripted Variables in the APT Language

The APT processor provides the capability of defining an array of geometric quantities or scalars that may

be referenced by one symbolic name. The individual quantities may be defined or referenced by means of a

subscript or an index.

A(1) = 2

A(2) = 5

The scalar quantity 2 is stored in the first storage location for A, and the scalar quantity 5 is stored in the

second storage location of A.

2.2.1. The Reserve (RESERV) Statement

When a symbol is used to define an array of quantities, the RESERV statement must be used to define the

extent of the array before any referencing to the subscripted symbol in the part program. For example:

RESERV/ A, 50

$$ or

RESERV/ 50, A

indicates that 50 storage locations are to be set aside for the symbol A. The first pair of entries in the

RESERV statement defines the order for subsequent pairs (symbol, range or range, symbol). The array(s)

defined in a RESERV statement may be used to store either scalars or surfaces. Statement ID's and MACRO

names cannot be subscripted. The first usage of a subscripted variable will determine whether surfaces (or

scalars) are to be allowed for that array. For example:

RESERV/ L1, 10, C1, 20, P1, 5, Q, 20

L1(2) = LINE/ 2.0, 1.0, 2.0, 5.0

C1(5) = CIRCLE/ 0, 0, 0, 0

P1(1) = POINT/ 0, 1, 0

Q(1) = 6.0 * 3 ** 2

would allow ten surfaces to be defined using the symbol L1(1)-L1(10), 20 surfaces to be defined using the

symbols C1(1)-C1(20), etc. Finally, 20 scalars could be defined using the symbols Q(1)-Q(20).

The extent of the array must be specified by a scalar number; a variable symbol cannot be used in place of

the required value.

$$ OK

RESERV/ A, 50

$$ Not allowed

B = 50

RESERV/ A, B

The value of the index (subscript) in a later part program statement cannot extend beyond the range assigned

to the symbol by the RESERV statement. The above example indicates that the allowable range for the

subscript is between 1 and 50, inclusive. Thus, the following examples would result in an error condition.

A(0) = 6

A(61) = 3.56

A(-1) = 10

The value of the subscript may be expressed as a number (as shown in the examples), as a symbolic scalar,

or as a computed expression. Although the subscript value may -- as a result, for instance, of having been

computed -- have a fractional part (that is, not be a perfect integer), it is truncated to an integer value when

used for subscripting purposes.

2.2.2. Inclusive Subscripts

Several input formats used in APT part programming call for writing strings (sometimes long) of data items

of the same class, such as all point definitions. Sometimes these data items have been calculated earlier in

the part program and can be defined as elements of a subscripted array. In place of writing a string of data of

the form:

A(1), A(2), A(3), A(4), ... A(n)

a modification of the subscript format permits the writing of a single item:

A(1, THRU, n)

which is exactly equivalent to the list of elements of the subscripted array A shown above.

The general form of this "inclusive subscript" notation is:

where a, b and c are scalar values expressed in any of the forms available in the APT language. This notation

is equivalent to a list of consecutive individual elements of the subscript array for the symbol NAME,

between subscript value a and subscript value b at interval of c. The value a specifies the lowest subscript of

the implied list, and be the highest value. These are not necessarily the first and last values of the entire

array; they must, of course, be within the range of the array.

the implied list is from a to b if the incremental value c is positive, and from be to a if the increment is

negative. A negative c may be specified by either a negative sign for the increment with the INCR modifier,

or a positive c with the DECR modifier. A negative c with DECR is equivalent to a double negative, the

effect being positive. The incremental modifier and value need not be given; if omitted, a positive increment

of 1 is assumed. When using incremental values other than 1, the final subscript of the implied list will be

the last subscript value that does not exceed the specified limit and need not be the same as the limiting

value. For Example:

A(1, THRU, 8, INCR, 3)

implies the list:

A(1), A(4), A(7)

When the value of the last subscript of the array is desired for b in the inclusive subscript notation, the

modifier ALL may be used, in place of a scalar following THRU, to indicate it. When the inclusive subscript

is to represent a list with the subscript limits of 1 and the highest value of the array, regardless of the

increment value, of direction, the notation "a, THRU, b" in the subscript nest may be replaced by the single

modifier ALL; for example:

A(ALL) or

A(ALL, DECR, 2)

The first element of the examples implies a list of all of the elements of the array A, in the order:

A(1), A(2), A(3), ... A(n)

The second example implies a list of every other element of the array A, starting with the last one, in the

order:

A(n), A(n-2), A(n-4), ...

If the low end of the range is to be 1, the value of a may be omitted. For example:

A(THRU, ALL)

is equivalent to:

A(1, THRU, ALL)

Examples of the use of inclusive subscripts:

L1 = LINE/ A(ALL)

where A is a scalar array with four or six elements.

L1 = LINE/ B(7, THRU, 10)

where B is a scalar array with ten or more elements.

L1 = LINE/ B(7, THRU, ALL)

where B is a scalar array with exactly ten or exactly twelve elements.

T1 = TABCYL/ NOZ, SPLINE, P(ALL)

where P is an array of three of more points.

In the general form:

NAME(a, THRU, b)

a should be the lower value of the range, and b should be the upper value of the range. If a=b,

P(5, THRU, 3)

has an implied positive direction and an increment value of 1. It should therefore define the low end of the

array P(5) and continue defining with a positive increment of 1 (P(6), P(7), ...) until the high end is reached

or passed. In this case, the high end, 3, is passed immediately, so P(6), etc., are not defined. However, the

one value P(5) is defined.

The converse example:

P(5, THRU, 3, DECR, 1)

is to start at the high end P(3) and decrease by 1 until the low end P(5) is reached. In this case, only P(3) will

be defined.

Inclusive subscript notation may be used in any geometric definition statement (with the one exception

noted in Section 3.22.2) and the PRINT or PUNCH statements. These are the only types of statements in

which the inclusive subscript is allowed.

Chapter 3. Geometric Statements in APT

The APT language permits the definition of a variety of geometric entities, with the added capability that

each of the allowable types can be defined in several ways. For instance, a point can be defined simply by

listing the x, y and z coordinates, or it can be defined as the intersection of two lines, or as one of the two

possible points of intersection of a line and a circle, etc. The geometric types allowed, together with the APT

vocabulary word used to indicate each, are listed below.

Table 3.1. Geometry Type and Name

Geometric Type Vocabulary Word

Point POINT

Line LINE

Plane PLANE

Circle CIRCLE

Cylinder CYLNDR

Ellipse ELLIPS

Hyperbola HYPERB

Cone CONE

General Conic GCONIC

Loft Conic LCONIC

Vector VECTOR

Matrix MATRIX

Sphere SPHERE

Quadric QADRIC

Tabulated Cylinder TABCYL

Polyconic Surface POLCON

Ruled Surface RLDSRF

3.1. Format

The general format of a surface definition is:

SNAME = SURFACE TYPE/ method of definition

where SNAME is a symbol, the SURFACE TYPE is one of the vocabulary words defining the surfaces

listed above and the method of definition is as defined in the following sections. A variable symbol may be

redefined, and then only as a surface of the same type, if one of two techniques involving the word CANON

is used.

CANON as a modifier

Any geometric definitions type involving the word CANON will allow the symbol to be redefined

(see Section 3.22).

These statements are provided to set a module condition to allow redefinition of surface variables.

The specification of CANON/ ON will allow subsequent surface variables to be redefined with any

geometric definition. The specification of CANON/ OFF, which is the default mode, will require the

CANON modifier for redefinition.

3.1.1. Modifiers

Some of the methods of definition require a modifier (or selector). This is necessary when there is more than

one solution as represented by the definitions itself. For example:

SPT = POINT/ XSMALL, INTOF, sline, scircle (see Figure 3.1)

would mean "define SPT as the point that has the smallest X coordinate value of the two possibilities when

the line sline is intersected with the circle scircle".

Figure 3.1. Geometry Modifier

3.1.2. Canonical Form

A particular geometric type can be defined in many ways. However, it would be highly inefficient to pass all

of this information along from section to section of the APT processor. Therefore, the APT processor

reduces to a common format each of the possible definitions for a particular geometric type. This format is

called the canonical form and is usually the most basic analytical expression for the surface type in question;

(Chapter 16 contains a table of the canonical forms for all of the allowable surface types.) For example, the

statement given in Section 3.1.1 would be evaluated, and only the x, y and z coordinates for the point would

be stored for later use.

3.2. The ZSURF Definition

The ZSURF definition supplies a means of defining an implicit Z coordinate for a point whenever it is

required for use in a geometric definition or a motion command. The general form of this statement is:

When only the X and Y coordinates for a point have been specified in a part program statement, the APT

processor projects the point on the ZSURF plane to calculate the implicit Z value. (if ZSURF is not

specified, the XY plane is assumed.) For example:

ZSURF/ SPLANE

SPOINT = POINT/ X, Y

The Z value is supplied by projecting the point (X, Y) in a direction parallel to the Z axis onto the plane

SPLANE.

3.3. The Point (POINT) Definitions

A point is a unique position in space and can be defined in several ways.

3.3.1. Rectangular Coordinates

SPT = POINT/ xcoord, ycoord, zcoord

SPT = POINT/ xcoord, ycoord

Figure 3.2. POINT by Rectangular Coordinates

PT1 = POINT/ 2, 4, 3

PT2 = POINT/ 2, 4, 0

PT3 = POINT/ -5, -2, 3

PT4 = POINT/ -5, -2, 0

3.3.2. Intersection of Two Lines

SPT = POINT/ INTOF, sline, sline

Figure 3.3. POINT by Line, Line Intersection

PT1 = POINT/ INTOF, LN1, LN4





3.3.3. Intersection of a Line and a Circle

Figure 3.4. POINT by Line, Circle Intersection

P1 = POINT/ YSMALL, INTOF, L1, C3

P2 = POINT/ YLARGE, INTOF, L1, C3

P3 = POINT/ YSMALL, INTOF, L1, C1

P4 = POINT/ YLARGE, INTOF, L1, C1

P5 = POINT/ XLARGE, INTOF, L2, C1

P6 = POINT/ XSMALL, INTOF, L2, C1



3.3.4. Intersection of Two Circles

Figure 3.5. POINT by Circle, Circle Intersection

P1 = POINT/ YLARGE, INTOF, C1, C2

P2 = POINT/ YSMALL, INTOF, C1, C2

P3 = POINT/ XSMALL, INTOF, C3, C4

P4 = POINT/ XLARGE, INTOF, C3, C4

3.3.5. On a Circle at an Angle with the X Axis

SPT = POINT/ scircle, ATANGL, degrees

The point lies on the circle so that the line from the point to the center of the circle makes a stated angle with

the X axis.

Figure 3.6. POINT by Circle, Angle

P1 = POINT/ C1, ATANGL, 45.0

P2 = POINT/ C1, ATANGL, 90

P3 = POINT/ C1, ATANGL, 225

$$ or

P1 = POINT/ C1, ATANGL, -315



Note

The angle is expressed in degrees and decimal fractions of a degree and is measured from the positive X

axis. A positive angle is measured counter clockwise.

3.3.6. Center of a Circle

SPT = POINT/ CENTER, scircle

Figure 3.7. POINT by Circle Center

PT1 = POINT/ CENTER, C1

3.3.7. Intersection of a Line and a Conic

Figure 3.8. POINT by Line, Conic Intersection



3.3.8. Intersection of Three Planes

SPT = POINT/ INTOF, splane, splane, splane

Figure 3.9. POINT by 3 Plane Intersection

P1 = POINT/ INTOF, PLANE1, PLANE2, PLANE3

3.3.9. Polar Coordinates

Figure 3.10. POINT by Polar Coordinates

PT1 = POINT/ RTHETA, XYPLAN, 5.66, 45

PT2 = POINT/ THETAR, XYPLAN, 150.0, 6

3.3.10. Intersection of a Line and a Tabulated Cylinder

SPT = POINT/ INTOF, sline, STABCYL, spoint

It is necessary to include a point on the TABCYL near the desired intersection to differentiate between the

several possible intersections. Further, since the APT processor uses this point as a basis for locating the

proper interval, a careful selection of this point can reduce processing time. The optimum point specification

is the last point of the TABCYL definition before the intersection point (that is, if the intersection point is

between the ninth and tenth points of the original TABCYL definition, using the ninth point as the "near

point" results in the most efficient processing). This is true for all the definitions involving a TABCYL

where a "near point" is required.

Figure 3.11. POINT by Line, Tabulated Cylinder Intersection

PNT = POINT/ INTOF, LN1, TBCYL1, PNTA

3.3.11. The Location in a PATERN

SPT = POINT/ spattern, n

Where n specifies the nth

point in the definition order of the pattern.

3.4. The Line (LINE) Definitions

A line is the intersection of two planes and is treated by the APT processor as a vertical plane surface (that

is, perpendicular to the XY plane).

3.4.1. Through Two Points

Note

The two points must not be coincidental

Figure 3.12. LINE by Two Points

LN1 = LINE/ -3, -3.5, 4, 3.5

LN2 = LINE/ 7, 2, 3, 1

$$ or

PT1 = POINT/ -3, -3.5

PT2 = POINT/ 4, 3.5

PT3 = POINT/ 7, 2

PT4 = POINT/ 3, 1

LN1 = LINE/ PT1, PT2

LN2 = LINE/ PT3, PT4

3.4.2. Through a Point and Tangent to a Circle

Note

The modifiers RIGHT and LEFT are applied looking from the point toward the circle.

Figure 3.13. LINE by Point and Tangent Circle

LN1 = LINE/ PT1, LEFT, TANTO, C1

LN2 = LINE/ PT1, RIGHT, TANTO, C1

LN3 = LINE/ PT2, RIGHT, TANTO, C1

3.4.3. Tangent to Two Circles

Note

The modifiers RIGHT and LEFT are applied looking from the first referenced circle toward the second.

Figure 3.14. LINE by Tangent Circles

L1 = LINE/ RIGHT, TANTO, C1, RIGHT, TANTO, C2

L2 = LINE/ LEFT, TANTO, C1, LEFT, TANTO, C2

L3 = LINE/ RIGHT, TANTO, C1, LEFT, TANTO, C2

L4 = LINE/ LEFT, TANTO, C1, RIGHT, TANTO, C2

3.4.4. Through a Point and at an Angle with the X or Y Axis

Figure 3.15. LINE by Point Angle

LN1 = LINE/ PT1, ATANGL, 45, XAXIS

LN2 = LINE/ PT2, ATANGL, 22.5

LN3 = LINE/ PT3, ATANGL, 60, YAXIS

3.4.5. Through a Point and Having a Slope with Respect to the X or Y Axis

The three lines illustrated in Figure 3.15 could be defined as follows:

LN1 = LINE/ PT1, SLOPE, 1, XAXIS

LN2 = LINE/ PT2, SLOPE, .4142

LN3 = LINE/ PT3, SLOPE, 1.7321, YAXIS

3.4.6. The X or Y Axis

3.4.7. Through a Point and Having a Slope with Respect to Another Line

SLN = LINE/ spoint, SLOPE, slope, sline

The lines L1 and L3 illustrated in Figure 3.16 could be defined as follows:

LN1 = LINE/ PT2, SLOPE, .26759, L2

LN3 = LINE/ PT3, SLOPE, -1, L1

3.4.8. Through a Point at an Angle with a Line

SLN = LINE/ spoint, ATANGL, degrees, sline

Note

A positive angle is measured counter clockwise from the given line. A negative angle is measured

clockwise.

Figure 3.16. LINE by Point, Angle to Line

L1 = LINE/ PT2, ATANGL, 15, L2

L3 = LINE/ PT3, ATANGL, 135, L1

3.4.9. Through a Point and Parallel to a Line

SLN = LINE/ spoint, PARLEL, sline

Figure 3.17. LINE by Point, Parallel Line

L1 = LINE/ PT1, PARLEL, LX

3.4.10. Through a Point and Perpendicular to a Line

SLN = LINE/ spoint, PERPTO, sline

Figure 3.18. LINE by Point, Perpendicular Line

LN1 = LINE/ PT1, PERPTO, LNX

3.4.11. Parallel to a Line at a Distance

Note

The modifiers denote the direction in which the line is offset

Figure 3.19. LINE by Distance from Parallel Line

L1 = LINE/ PARLEL, LX, XSMALL, 2

$$ or

L1 = LINE/ PARLEL, LX, YLARGE, 2.0

L2 = LINE/ PARLEL, LX, XLARGE, 2

$$ or

L2 = LINE/ PARLEL, LX, YSMALL, 2.0

3.4.12. Intersection of Two Planes

SLN = LINE/ INTOF, splane, splane

Note

The planes cannot be coplanar, parallel to one another, or both vertical (perpendicular to the XY plane).

Figure 3.20. LINE by Plane Intersection

L1 = LINE/ INTOF, PLN1, PLN2

3.4.13. Slope Intercept Form

Figure 3.21. LINE by Slope Intercept Form

L1 = LINE/ SLOPE, .40403, INTERC, 3

L2 = LINE/ SLOPE, -.40403, INTERC, 3

$$ or

L1 = LINE/ SLOPE, .40403, INTERC, YAXIS, 3

L2 = LINE/ SLOPE, -.40403, INTERC, XAXIS, 7.425

$$ or

L1 = LINE/ ATANGL, 22, INTERC, 3

L2 = LINE/ ATANGL, -22, INTERC, 3

$$ or

L1 = LINE/ ATANGL, 22, INTERC, YAXIS, 3

L2 = LINE/ ATANGL, -22, INTERC, XAXIS, 7.425

3.4.14. Angle and Axis Intercept

See Figure 3.21

3.4.15. Through a Point and Tangent to a Tabulated Cylinder

SLN = LINE/ spoint, TANTO, stabcyl, spoint

Note

A point on the TABCYL near the point of tangency is included to specify the desired point of tangency.

(This point may be omitted if it is the same as the spoint specified immediately after the slash.)

Figure 3.22. LINE by Point, Tangent Tabulated Cylinder

L3 = LINE/ P3, TANTO, TABC, P4

3.4.16. Through a Point and Perpendicular to a Tabulated Cylinder

SLN = LINE/ spoint, PERPTO, stabcyl, spoint

Note

A point on the TABCYL near the normal is included to specify the desired normal.

Figure 3.23. LINE by Point, Tangent Tabulated Cylinder

L2 = LINE/ P1, PERPTO, TABC, P2

3.5. The Plane (PLANE) Definitions

A plane is a surface that contains all points of a straight line joining any two points of the surface.

3.5.1. Coefficients of the Plane Equation

SPL = PLANE/ a, b, c, d

Figure 3.24. PLANE by Coefficients

PLANA = PLANE/ 0, 0, 1, 1.5

PLANB = PLANE/ .5, 0, 1, -4

3.5.2. Passing Through Three Nonlinear Points

SPL = PLANE/ spoint, spoint, spoint

Figure 3.25. PLANE by Three Points

P1 = POINT/ 1, 1, 1

P2 = POINT/ 2, 3, 2

P3 = POINT/ 3, 4, 0

PLN = PLANE/ P1, P2, P3

3.5.3. Through a Point and Parallel to a Plane

SPL = PLANE/ spoint, PARLEL, splane

Figure 3.26. PLANE by Point, Parallel Plane

PL1 = PLANE/ P1, PARLEL, PLN

PL2 = PLANE/ P2, PARLEL, PLN

3.5.4. Parallel to a Plane at a Distance

Figure 3.27. PLANE by Parallel Plane at Distance

PLQ = PLANE/ PARLEL, PL5, XLARGE, D

3.5.5. Through a Point and Perpendicular to a Vector

SPL = PLANE/ spoint, PERPTO, svector

Figure 3.28. PLANE by Point, Vector

PLN1 = PLANE/ PT, PERPTO, VEC

3.5.6. Through Two Points and Perpendicular to a Plane

Note

A line joining the two given points must not be perpendicular to the given plane.

Figure 3.29. PLANE by Two Points, Perpendicular Plane

PLN6 = PLANE/ PERPTO, PL2, PTA, PTB

3.5.7. Perpendicular to Two Intersecting Planes and Passing Through a Point

SPL = PLANE/ spoint, PERPTO, splane, splane

3.6. The Circle (CIRCLE) Definitions

A circle is the locus of points that are equidistant from a fixed point. The APT processor defines a circle as a

cylinder perpendicular to the XY plane.

3.6.1. Coordinates of the Center and the Radius

SCIR = CIRCLE/ xcoord, ycoord, zcoord, radius

SCIR = CIRCLE/ xcoord, ycoord, radius

Figure 3.30. CIRCLE by Coordinates, Radius

C1 = CIRCLE/ 4, 3, 2

$$ or

PT1 = POINT/ 4, 3

C1 = CIRCLE/ CENTER, PT1, RADIUS, 2

3.6.2. Center and a Line to which it is Tangent

SCIR = CIRCLE/ CENTER, spoint, TANTO, sline

Figure 3.31. CIRCLE by Center Point, Tangent Line

C1 = CIRCLE/ CENTER, PT2, TANTO, L1

3.6.3. Center and a Point on the Circumference

SCIR = CIRCLE/ CENTER, spoint, spoint

Figure 3.32. CIRCLE by Center Point, Circumference Point

C1 = CIRCLE/ CENTER, PNTB, PNTC

3.6.4. Three Points on the Circumference

SCIR = CIRCLE/ spoint, spoint, spoint

Figure 3.33. CIRCLE by Three Points

C1 = CIRCLE/ PT1, PT2, PT3

3.6.5. Center Point and Tangent to a Circle

Note

There are two possibilities. The modifiers LARGE and SMALL indicate the circle is to be chosed with

the largest or smallest (respectively) radius.

Figure 3.34. CIRCLE by Center Point, Tangent Circle

C2 = CIRCLE/ CENTER, PT1, SMALL, TANTO, C1

C3 = CIRCLE/ CENTER, PT1, LARGE, TANTO, C1

3.6.6. Radius and Tangent to Two Intersecting Lines

Figure 3.35. CIRCLE by Radius, Two Tangent Lines

C1 = CIRCLE/ YLARGE, L2, YLARGE, L1, RADIUS, .375

C2 = CIRCLE/ YSMALL, L1, XLARGE, L2, RADIUS, .375

C3 = CIRCLE/ YSMALL, L2, YSMALL, L1, RADIUS, .375

C4 = CIRCLE/ XSMALL, L2, YLARGE, L1, RADIUS, .375

$$ or

C1 = CIRCLE/ XLARGE, L2, XSMALL, L1, RADIUS, .375

C2 = CIRCLE/ XLARGE, L1, YLARGE, L2, RADIUS, .375

C3 = CIRCLE/ XSMALL, L2, XLARGE, L1, RADIUS, .375

C4 = CIRCLE/ YSMALL, L2, XSMALL, L1, RADIUS, .375

3.6.7. Radius and Tangent to a Line and Passing Through a Point

Figure 3.36. CIRCLE by Radius, Point, Tangent Line

C1 = CIRCLE/ TANTO, LN1, XSMALL, PT1, RADIUS, .5

C2 = CIRCLE/ TANTO, LN1, XLARGE, PT1, RADIUS, .5

$$ or

C1 = CIRCLE/ TANTO, LN1, YSMALL, PT1, RADIUS, .5

C2 = CIRCLE/ TANTO, LN1, YLARGE, PT1, RADIUS, .5

3.6.8. Radius and Tangent to a Line and a Circle

Figure 3.37. CIRCLE by Radius, Tangent Line, Tangent Circle

C1 = CIRCLE/ YLARGE, LG, XSMALL, OUT, CG, RADIUS, 1.

C2 = CIRCLE/ YLARGE, LG, XSMALL, IN, CG, RADIUS, 1.

C3 = CIRCLE/ YSMALL, LG, XSMALL, OUT, CG, RADIUS, 1.

C4 = CIRCLE/ YSMALL, LG, XSMALL, IN, CG, RADIUS, 1.

C5 = CIRCLE/ YLARGE, LG, XLARGE, IN, CG, RADIUS, 1.

C6 = CIRCLE/ YSMALL, LG, XLARGE, IN, CG, RADIUS, 1.

C7 = CIRCLE/ YLARGE, LG, XLARGE, OUT, CG, RADIUS, 1.

C8 = CIRCLE/ YSMALL, LG, XLARGE, OUT, CG, RADIUS, 1.

3.6.9. Radius and Tangent to Two Circles

Figure 3.38. CIRCLE by Radius, Two Tangent Circles

CIR3 = CIRCLE/ YLARGE, IN, CIR2, OUT, CIR1, RADIUS, .25

CIR5 = CIRCLE/ YSMALL, OUT, CIR2, OUT, CIR4, RADIUS, .5

CIR6 = CIRCLE/ YLARGE, OUT, CIR2, OUT, CIR4, RADIUS, .5

3.6.10. Radius and Tangent to a Line and a Tabulated Cylinder

Note

A point on the TABCYL must be included that is nearest the desired tangency point.

Figure 3.39. CIRCLE by Radius, Tangent Line, Tangent TABCLY

CL1 = CIRCLE/ TANTO, L1, XSMALL, TABC, YLARGE, PNTA, RADIUS, .5

3.7. The Cylinder (CYLNDR) Definitions

A cylinder is the locus of all points at a constant distance from a given line.

3.7.1. An Axis Point, an Axis Vector and a Radius

SCYL = CYLNDR/ x, y, z, a, b, c, radius

Note

(x, y, z) represent a point on the axis of the cylinder, (a, b, c) are the components of the unit vector in

the direction of the axis and (radius) represents the radius of the cylinder.

Figure 3.40. CYLNDR by Point, Vector, Radius

C1 = CYLNDR/ 8, 0, 1, 0, 0, 1, 1.0

3.7.2. Substitution of Symbols in the Canonical Form

SCYL = CYLNDR/ spoint, a, b, c, radius

SCYL = CYLNDR/ x, y, z, svector, radius

SCYL = CYLNDR/ spoint, svector, radius

For example, the cylinder in Figure 3.40 could also be defined:

C1 = CYLNDR/ PTA, 0, 0, 1, 1.0

C1 = CYLNDR/ 8, 0, 1, VECTA, 1.0

C1 = CYLNDR/ PTA, VECTA, 1.0

3.8. The Ellipse (ELLIPS) Definition

An ellipse is a plane curve that is the locus of a point moving in such a manner that the sum of the distances

from that point to two fixed points in the plane is constant.

3.8.1. Center, Semi-Major and Semi-Minor Axes, and the Angle the Major Axis Makes

with the X Axis.

SELP = ELLIPS/ CENTER, spoint, semi-maj, semi-min, angle

Note

The ellipse define will be in the XY plane. A positive angle (in degrees) is measured counter clockwise.

Figure 3.41. ELLIPS Example

EL1 = ELLIPS/ CENTER, PTA, 3.25, 1.9, 30

3.9. The Hyperbola (HYPERB) Definition

A hyperbola is a plane curve that is the locus of a point moving such that the differences of the distances

from that point to two fixed points in the plane are consistant.

3.9.1. Center, Half-Tranverse Axis, Half-Conjugate Axis and the Angle of the Transverse

Axis to the X axis.

SHYP = HYPERB/ CENTER, spoint, half-transverse, half-conjugate, angle

Note

A positive angle (in degrees) is measured counter clockwise.

Figure 3.42. HYPERB Example

HYP1 = HYPERB/ CENTER, PTA, 2, 1, 45

3.10. The Cone (CONE) Definitions

A cone is a surface generated by moving a straight line so that the line always touches a circle, and passes

through a fixed point.

3.10.1. Canonical Form

SCN = CONE/ CANON, x, y, z, a, b, c, costheta

Note

(x, y, z) are the coordinates of the vertex of the cone, (a, b, c) are the components of the unit vector in

the direction of the axis of the cone, and (costheta) is the cosine of the vertex half-angle.

Figure 3.43. CONE by Canonical Form

CONEA = CONE/ CANON, 3.1, 2, 0, 0, 0, 1, .866

3.10.2. Expressing the Vertex and Axis Vector Symbolically

SCN = CONE/ spoint, svector, half-angle

For example, the cone in Figure 3.43 could also be defined as:

CONEA = CONE/ PTA, VECB, 30.0

3.11. The General Conic (GCONIC) Definitions

The general conic is the plane curve represented by the general class of second-degree equations in two

variables:

Ax2 + Bxy + Cy

2 + Dx + Ey + F = 0

3.11.1. Coefficients of the General Equation

SGC = GCONIC/ A, B, C, D, E, F

3.11.2. Coefficients of the Alternate Equation

SGC = GCONIC/ P, Q, R, S, T

Where the alternate equation is defined as follows:

3.11.3. Coefficients of the Inverse of the Alternate Equation

SGC = GCONIC/ P, Q, R, S, T, FUNOFY

Where the alternate equation is defined as follows:

3.12. The Loft Conic (LCONIC) Definitions

A loft conic is a general conic expressed by a combination of five independent conditions, rather than by a

mathematical expression.

3.12.1. Five Points

Figure 3.44. LCONIC by Five Points

LCONA = LCONIC/ SPT, PTA, PTB, PTC, PTD, PTE

3.12.2. Four Points and a Slope

Figure 3.45. LCONIC by Four Points, Slope

LC1 = LCONIC/ 4PT1SL, PTA, 1.0, PTB, PTC, PTD

3.12.3. Three Points and a Two Slopes

Figure 3.46. LCONIC by Three Points, Two Slopes

LCOND = LCONIC/ 3PT2SL, PTA, 1.887, PTB, -.577, PTC

3.13. The Vector (VECTOR) Definitions

The vector is that quantity which has both magnitude and direction. In cases where a specific magnitude is

not defined for a vector, the unit vector (magnitude = 1) is assumed.

The formats given in Section 3.13.4, Section 3.13.5, Section 3.13.6 and Section 3.13.9 accept as input both

points and vectors. In these formats, if a point is specified it is equivalent to a vector from the origin to the

given point.

3.13.1. X, Y, Z Components

SVCT = VECTOR/ x, y, z

Figure 3.47. VECTOR by X, Y, Z

VEC1 = VECTOR/ 1, 1, 4

VEC2 = VECTOR/ +2, -6, -1

$$ or

VEC1 = VECTOR/ 2, 2, -3, 3, 3, 1

VEC2 = VECTOR/ -5, 2, 3, -3, -4, 2

$$ or

VEC1 = VECTOR/ P1, P2

VEC2 = VECTOR/ P4, P3

3.13.2. Two End Points

Note

The defined vector is from the first point toward the second point. (See Figure 3.47)

3.13.3. Perpendicular to a Plane

The modifiers are used to indicate the direction the vector is to point. (POSX indicates the vector points in

the positive X direction, etc.)

Figure 3.48. VECTOR by Perpendicular Plane

VECTA = VECTOR/ PERPTO, PLANA, POSX

3.13.4. A Scalar Times a Vector

SVCT = VECTOR/ scalar, TIMES, svector

Figure 3.49. VECTOR by Scaled Vector

VEC3 = VECTOR/ 4.0, TIMES, VEC

VEC4 = VECTOR/ -1, TIMES, VEC

3.13.5. The Cross Product of Two Vectors or Points

SVCT = VECTOR/ svector, CROSS, svector

Note

The resultant vector will be perpendicular to the plane of the two given vectors, and its length will be

the scalar product of the magnitudes of the given vectors and the sine of the included angle. The

direction of the resultant vector will be as see in Figure 3.50 (an application of the "right-hand rule").

Figure 3.50. VECTOR by Cross Product

VECTA = VECTOR/ V1, CROSS, V2

VECTB = VECTOR/ V2, CROSS, V1

3.13.6. Normalizing a Vector, Point or Components

Note

A normalized vector has the same direction as the given vector, with a magnitude equal to one.

Figure 3.51. VECTOR by Normal

VECTA = VECTOR/ UNIT, VECB

3.13.7. A Length (Magnitude) and an Angle in a Plane

The angle is measured from the first axis specified in the modifier; that is, from the X axis in the XYPLAN,

from the Y axis in the YZPLAN, and from the Z axis in the ZXPLAN.

Figure 3.52. VECTOR by Length, Angle

VEC = VECTOR/ LENGTH, 6, ATANGL, 30, XYPLAN

3.13.8. Parallel to the Intersection of Two Planes

Figure 3.53. VECTOR by Intersecting Planes

VECA = VECTOR/ PARLEL, INTOF, PLNA, PLNB, NEGZ

3.13.9. The Addition or Subtraction of Two Vectors or Points

SVCT = VECTOR/ svector, PLUS, svector

SVCT = VECTOR/ spoint, PLUS, spoint

SVCT = VECTOR/ svector, MINUS, svector

SVCT = VECTOR/ spoint, MINUS, spoint

Figure 3.54. VECTOR by Addition, Subtraction

VECTC = VECTOR/ V1, PLUS, V2

VECTD = VECTOR/ V1, MINUS, V2

3.13.10. In the XY Plane Having an Angle with a Line

Figure 3.55. VECTOR by Angle, Line

V1 = VECTOR/ ATANGL, 30, L1, POSY

V1 = VECTOR/ ATANGL, 30, L1, XLARGE

V2 = VECTOR/ ATANGL, 135, L1, XSMALL

V2 = VECTOR/ ATANGL, 135, L1, NEGX

V3 = VECTOR/ ATANGL, 330, L1, POSX

V3 = VECTOR/ ATANGL, 330, L1, XLARGE

3.14. The Matrix (MATRIX) Definitions

A matrix is a rectangular array of numbers defining the three simultaneous equations that represent the

relationship between two coordinate systems.

3.14.1. The Equation Coefficients (Canonical Form)

SMAT1 = MATRIX/ a1, b1, c1, d1, a2, b2, c2, d2, a3, b3, c3, d3

where a1, b1, etc. are the coefficients of the equations:

a1x + b1y + c1z - d1 = 0

a2x + b2y + c2z - d2 = 0

a3x + b3y + c3z - d3 = 0

which define three mutually perpendicular planes of the new coordinate system.

3.14.2. A Translation of Axis

SMAT2 = MATRIX/ TRANSL, d1, d2, d3

Note

d3 may be omitted. If it is omitted, the APT processor assumes d3 = 0.

The parameters d1, d2 and d3 are the coordinates of the origin of the new coordinate system expressed in

terms of the old system.

The format will generate the following canonical form:

1, 0, 0, d1, 0, 1, 0, d2, 0, 0, 1, d3

3.14.3. A Rotation of Axis

Note

angle is in degrees

a. Rotation in the XY plane, where positive rotation is from the X axis to the Y axis (counter clockwise)

M1 = MATRIX/ XYROT, A

The canonical form generally will be:

COS A, -SIN A, 0, 0, SIN A, COS A, 0, 0, 0, 0, 1, 0

b. Rotation in the YZ plane, where positive rotation is from the Y axis to the Z axis (counter clockwise)

M2 = MATRIX/ YZROT, B


1, 0, 0, 0, 0, COS B, -SIN B, 0, 0, SIN B, COS B, 0

c. Rotation in the ZX plane, where positive rotation is from the Z axis to the X axis (counter clockwise)

M3 = MATRIX/ ZXROT, C


COS C, 0, SIN C, 0, 0, 1, 0, 0, -SIN C, 0, COS C, 0

3.14.4. Matrix Combinations

As shown in Section 3.14.5 a MATRIX can be defined as a combination (Multiplication) of matrices.

Because matrix operations of this type are not commutative, the order of the operation becomes quite

important. That is, if M and N are matrices, in general:

M x N N x M

To illustrate, compare the result of performing a rotation followed by a translation with the result obtained

by first performing the translation and then the rotation.

If matrix A is multiplied into matrix B, the resultant matrix C will have the effect of first performing the B

operations and then performing the A operations. For example, let A be a translation matrix and B be a

rotation matrix. Then:

Atranslation x Brotation = C

Matrix C will have the effect of first a rotation and then a translation. On the other hand, if the matrix

multiplication is reversed:

Arotation x Btranslation = C

C will first translate and then rotate. Notice that the effect of a matrix multiplication can be thought of as

being the opposite of the order of multiplication.

3.14.5. The Product of Two Matrices

The order of matrix multiplication for most of the combinations above is left to right. For example:

M1 = MATRIX/ MATR, MATT

where MATR is a rotation matrix and MATT is a translation matrix will yield:

M1 = MATR x MATT

There are exceptions to this general rule. The formats given below define the cases in which the order of

matrix multiplication is right to left rather than left to right.

All other formats result in a matrix multiplication order of left to right.

Assume matrix is multiplied into matrix , where:

a41 = b41 = 0

a42 = b42 = 0

a43 = b43 = 0

a44 = b44 = 1

The canonical form generated will be:

Obviously, the order of the matrices given in this format is quite important.

3.14.6. The Inverse of a Matrix

SMAT6 = MATRIX/ INVERS, smatrix

If the input matrix is defined as:

a1, b1, c1, d1, a2, b2, c2, d2, a3, b3, c3, d3,

the canonical form generated will be:

A1 = a1

B1 = a2

C1 = a3

D1 = -(a1d1 + a2d2 + a3d3)

A2 = b1

B2 = b2

C2 = b3

D2 = -(b1d1 + b2d2 + b3d3)

A3 = c1

B3 = c2

C3 = c3

D3 = -(c1d1 + c2d2 + c3d3)

3.14.7. Three Mutually Perpendicular Planes

SMAT7 = MATRIX/ splane, splane, splane

If the canonical forms of the three planes are:

PLAN1: A1, B1, C1, D1

PLAN2: A2, B2, C2, D2

PLAN3: A3, B3, C3, D3

the canonical form generated will be:

A1, A2, A3, -(A1D1 = A2D2 + A3D3),

B1, B2, B3, -(B1D1 = B2D2 + B3D3),

C1, C2, C3, -(C1D1 = C2D2 + C3D3)

3.14.8. Scale Factor

SMAT8 = MATRIX/ SCALE, s

The canonical form generated will be:

s, 0, 0, 0, 0, s, 0, 0, 0, 0, s, 0

3.14.9. Origin, X Axis Vector and a Vector in the 1st or 2nd Quadrant

SMAT = MATRIX/ spoint, svector1, svector2

Where svector1 represents the new X axis and svector2 is in the first or second quadrant.

3.14.10. A Mirror Matrix about One or More Planes

3.14.11. A Mirror Matrix about a Line or Plane

3.15. The Sphere (SPHERE) Definitions

A sphere is a surface such that all points on the surface are equidistant from a fixed point.

3.15.1. Center Point and Radius

Figure 3.56. SPHERE by Center Point, Radius

S1 = SPHERE/ PNTC, 2.5

$$ or

S1 = SPHERE/ CENTER, PNTC, PNTA

3.15.2. Center Point and a Point on the Surface

SPH = SPHERE/ CENTER, spoint, spoint

See Figure 3.56.

3.15.3. Center Point and a Plane to which it is Tangent

SPH = SPHERE/ CENTER, spoint, TANTO, splane

Figure 3.57. SPHERE by Center Point, Tangent Plane

SPHA = SPHERE/ CENTER, PTA, TANTO, PLANEB

3.15.4. Passing Through Four Points

SPH = SPHERE/ spoint, spoint, spoint, spoint

Note

The four points cannot be coplanar.

Figure 3.58. SPHERE by Four Points

SPHD = SPHERE/ PNTA, PNTB, PNTC, PNTD

3.16. The Quadric (QADRIC) Definitions

The general qadric surface is the locus of points that satisfy the following general equation in the second

degree:

SQAD = QADRIC/ a, b, c, d, e, f, g, h, p, q, r, d

3.16.1. Permissible Quadric Surfaces

Figure 3.59 through Figure 3.62 and Figure 3.63 through Figure 3.67 show all allowable quadric surfaces,

together with the normal equation for each surface.

Figure 3.59. Elliptic Cone

Figure 3.60. Elliptic Cylinder

Figure 3.61. Hyperbolic Cylinder

Figure 3.62. Parabolic Cylinder

Figure 3.63. Real Ellipsoid

Figure 3.64. Hyperboloid of One Sheet

Figure 3.65. Hyperboloid of Two Sheets

Figure 3.66. Elliptic Paraboloid

Figure 3.67. Hyperboloid Paraboloid

3.16.2. Quadratics as a Special Form of QADRIC

In many instances a quadratic (that is, parabola, ellipse, hyperbola) is defined in equation form. For

example, an ellipse my be specified in the following form:

Ax2 + By

2 + Cx + Dy + E = 0

ELLP1 = QADRIC/ A, B, 0, 0, 0, 0, C, D, 0, E

As further illustration, the parabola (y - a)2

- p(x - b) = 0 can be expressed

y2 - 2ay + a

2 - px + pb = 0

or, collecting terms:

y2 - 2ay - px + (pb + a

2 )

The general quadric definition can then be used as follows:

PARB = QADRIC/ 0, 1, 0, 0, 0, 0, -p,$$

(-2 * a), 0, (p * b + a**2)

This provides the facility for specifying any two-dimensional quadratic curve with the QADRIC surface

definition.

3.15. The Sphere (SPHERE) Definitions

3.17. The Reference System (REFSYS)

Statement

3.17. The Reference System (REFSYS) Statement

Chapter 3. Geometric Statements in APT

3.17. The Reference System (REFSYS) Statement

The REFSYS statement is used to describe a surface or surfaces in one coordinate system (such as an

auxiliary view on a blueprint) and to machine this surface or surfaces in another coordinate system.

REFSYS/ smatrix

REFSYS/ NOMORE

The REFSYS statement specifies a matrix that will be used to transform the geometric surface or surfaces to

any other coordinate system. smatrix expresses the origin and coordinate planes of the new system in terms

of the old system (or base system). A particular reference system is in effect until it is either replaced by a

new reference system (REFSYS/ smatrix) or terminated by a REFSYS/ NOMORE, which causes a return to

the base or original coordinate system. All of the geometric types listed in Section 3.3 through Section 3.16,

except the MATRIX definition itself, can be transformed by using the REFSYS statement. No other APT

surface can be transformed. If a nested definition is used in the REFSYS statement, the MATRIX must be

given a name.

A geometric definition defined in a particular reference system will not be transformed until it is used in a

reference system different (by name) from the system in which it was defined. Example:

REFSYS/ (MAT1 = MATRIX/ TRANSL, 5, 5, 0)

P1 = POINT/ 2, 4, 0

REFSYS/ NOMORE

A) GOTO/ P1

The point P1 will be defined with the coordinate values X = 2, Y = 4, Z = 0. At statement labeled A, the

point P1 will be used to define the output point -- by applying the translation matrix MAT1, the output point

will have the coordinate values X = 7, Y = 9, Z = 0; P1 will still be defined as X = 2, Y = 4, Z = 0 in

reference system MAT1. To rephrase, a geometric definition when used in another system will be

dynamically redefined as often as it is used while preserving its static definition in its own system.

Note

When both rotation and translation are specified, the translation is made first, then the rotation.

3.18. The Tabulated Cylinder (TABCYL) Definition

A tabulated cylinder (TABCYL) is the surface generated by moving a line (generatrix) along a space curve

(directrix) such that it is always parallel to a given line. The space curve is defined by a set of points and an

interpolation scheme for "fairing" between given points. A maximum of 139 points may be given to define

the space curve. However, when using some of the formats available for TABCYL specification, it is

possible to exceed the limitation on the maximum number of elements in one statement (see Section 1.1.6)

well before the 139 point limit is reached. The limiting value in such cases depends upon the number of

elements being used in the description of each point, and must be determined by the part programmer.

The general format of the TABCYL is:

Note

A symbolic vector (svector) must be specified if, and only if, the XYZ format is used.

The various options available under the formats indicated above (CANON, NOX, NOY, NOZ, RTHETA,

THETAR, XYZ), together with the data representations, are discussed in the following sections.

3.18.1. Transformation (TRFORM)

A TABCYL can be transformed at the time it is defined by using TRFORM and specifying the symbolic

MATRIX (smatrix) to be used.

3.18.2. CANON Format

A TABCYL can be defined by specifying the canonical form of parameters:

STAB = TABCYL/ CANON, n, k, m1, ... m9, u1, v1, a1, b1, c1, r1, u2, v2, a2, b2, c2, r2, ... un, vn, an, bn, cn, rn,

un+1, vn+1,

See Section 16.15 for an explanation of the terms.

3.18.3. NOZ, RTHETA, THETAR Formats

The data in these formats is given in the XY plane, the generatrix is normal to the XY plane; and coordinate

transformation is limited to the XY plane. NOZ implies X, Y coordinates only are given; RTHETA and

THETAR indicate polar coordinates, where r is the radius (in units) and is the angle in degrees measured counter clockwise from the positive X axis.

Note

RTHETA means r1, 1, r2, 2, ... rn, n, and THETAR implies 1, r1, 2, ... r2, n, rn.

The following sections discuss the options (PTNORM, PTSLOP, TWOPT, FOURPT, SPLINE) available

under these formats.

Note

The NOZ format is used, although the options also apply to RTHETA and THETAR

3.18.3.1. PTNORM

STAB = TABCYL/ NOZ, PTNORM, [TRAFORM, smatrix,] x1, y1, n1, x2, y2, n2, ... xn, yn, nn

This option is used when the normals (n1, n2, ... nn, where n is specified in degrees measured counter

clockwise from the positive X axis) are to be input for each given point. The interpolation technique defines

a series of curves that go through the given points, matching the specified normals at each point.

3.18.3.2. PTSLOP

STAB = TABCYL/ NOZ, PTSLOP, [TRAFORM, smatrix,] x1, y1, s1, x2, y2, s2, ... xn, yn, sn

This option is used when the slopes (s1, s2, ... sn, where s is specified as y / x) are known at each given point. The fitting scheme is such that the slopes are matched at each point.

3.18.3.3. TWOPT

This option is used when the slopes or normals are specified only at selected points. The program computes

a slope or normal when not specified; it then processes as in Section 3.18.3.1 or Section 3.18.3.2.

3.18.3.4. FOURPT

STAB = TABCYL/ NOZ, FOURPT, [TRAFORM, smatrix,] x1, y1, x2, y2, ... xn, yn

This option defines a series of curves, each one passing through four successive input points. The resulting

TABCYL does not have a continuous slope.

3.18.3.5. SPLINE

This TABCYL is similar to the TWOPT, PTNORM and PTSLOP types, except that the slopes or normals

are adjusted until the curvature is continuous.

3.18.4. NOX, NOY Formats

These formats are identical to the NOZ format (Section 3.18.3) except that the data reference plane is

different.

NOX implies the XZ plane and requires that the points be specified in the y, z order (y1, z1, y2, z2, ..., yn,

zn). NOY implies the ZX plane and requires that the points be specified in the z, x order (z1, x1, z2, x2, ...,

zn, xn).

3.18.5. XYZ Format

This format requires that the symbolic vector (svector) representing the generatrix be specified. Further, the

processor determines which coordinate plane will control the fitting techniques; therefore, no slopes or

normals can be given.

The options (TWOPT, FOURPT, SPLINE) are as outlined in Section 3.18.3.3 through Section 3.18.3.5,

respectively, with the exception that no slopes are given in the TWOPT method.

3.19. The Polyconic (POLCON) Definition

The polyconic surface is defined to be a continuous family of conic sections in parallel planes. The shape of

the conics is regulated by a set of polynomial curves.

Figure 3.68. POLCON Nomenclature

3.19.1. Method of Definition

The input to the polyconic routine is the canonical form of the polyconic, as follows:

POLY1 = POLCON/ CANON, JROOTS,

M1, M2, M3, M4, M5, M6, M7, M8, M9, M10, M11, M12,

T, Cb, Pb,

A0, A1, A2, A3, A4, A5, A6, A7,

B0, B1, B2, B3, B4, B5, B6, B7,

C0, C1, C2, C3, C4, C5, C6, C7,

D0, D1, D2, D3, D4, D5, D6, D7,

L0, L1, L2, L3, L4, L5, L6, L7,

H0, H1, H2, H3, H4, H5, H6, H7,

P0, P1, P2, P3, P4, P5, P6, P7,

A 1/2, B 1/2, C 1/2, D 1/2, L 1/2, H 1/2, P 1/2

where:

JROOTS

Equal to 2 if there are square root terms, 1 otherwise

M1 - M12

Transformation matrix (may be all zeros)

T

Thickness between defined surface and working surface

Cb

Origin of curves, that is, the start of the polyconic

Pb

Full-size length of the valid region of the surface

A0-P1/2

Coefficients of surface control equations, each of the form:

f(d) = A 1/2 d + A0 + A1d + a2d2 + a3d3 + a4d4 + a5d5 + a6d6 + a7d7

Note

The coefficients of the square root terms for all the surface control equations are in the bottom line of

the canonical form. The square root terms may all be zero (JROOTS = 1).

See Figure 3.68 for a graphic representation of a POLCON.

3.20. The Ruled Surface (RLDSRF) Definitions

A ruled surface is a surface generated by straight lines (called rulings) joining corresponding points on two

space curves. An alternate form of the ruled surface allows substitution of a point of the second space curve.

The space curves are two-dimensional (formed by a plane cutting a surface) and are limited by two end

points. The line joining the end points of a space curve is a base line, whose length considered as unity, or

100 percent. A point on one space curve corresponds to a point on the second space curve when the

associated points on the base lines are of equal percentage.

The following surfaces are permitted within a ruled surface definition: line, plane, circle, cylinder, ellipse,

hyperbola, cone, general conic, loft conic, sphere, quadric and tabulated cylinder.

3.20.1. Six Points and Two Surfaces

A ruled surface may be defined by specifying:

A surface and three points on a plane intersecting the surface (two of the points being end points of the

desired space curve), and a second surface and three points on a plane intersecting the surface (two of the

points being the end points of the second space curve).

Figure 3.69. RLDSRF by Six Points, Two Surfaces

RS1 = RLDSRF/ SURF1, P1A, P1B, P1C, SURF2, P2A, P2B, P2C

Format:

SRLD = RLDSRF/ ssurface, spoint, spoint, spoint, ssurface, spoint, spoint, spoint

With reference to Figure 3.69, the following restrictions apply:

P1A, P1B and P1C cannot be colinear (similarly for P2A, P2B and P2C)

P1A, P1B and P2A, P2B must be the end points of the first and second space curves respectively.

3.20.2. Two Surfaces, Four Points and Two Vectors


A surface and two end points of a plane curve and a vector, and a second surface with two end points of a

space curve and a vector. (The two points and the vector determine an intersecting plane in each case.)

Figure 3.70. RLDSRF by Two Surfaces, Four Points, Two Vectors

RS2 = RLDSRF/ SURF1, P1A, P1B, V1, SURF2, P2A, P2B, V2

Format:

SRLD = RLDSRF/ ssurface, spoint, spoint, svector, ssurface, spoint, spoint, svector

3.20.3. Two Surfaces, Five Points and a Vector


Three non-colinear points to define one intersecting plane and two points, and a vector to define the other

intersecting plane.

Figure 3.71. RLDSRF by Two Surfaces, Five Points, Vector

RS3 = RLDSRF/ SURF1, P1A, P1B, V1, SURF2, P2A, P2B, P2C

$$ or

RS3 = RLDSRF/ SURF1, P1A, P1B, P1C, SURF2, P2A, P2B, V2

Format:

SRLD = RLDSRF/ ssurface, spoint, spoint, svector, ssurface, spoint, spoint, spoint

or

SRLD = RLDSRF/ ssurface, spoint, spoint, spoint, ssurface, spoint, spoint, svector

3.20.4. Replacing One of the Curves with a Point

A rules surface may be defined by using a point as one of the surfaces.

Figure 3.72. RLDSRF by Replacing Curve with Point

RS4 = RLDSRF/ SURF1, P1A, P1B, P1C, P2

$$ or

RS4 = RLDSRF/ SURF1, P1A, P1B, V1, P2

Format:

See Section 14.6 for further information on RLDSRF.

3.21. Nested Definitions

The nested definition permits defining a geometric quantity (except a scalar variable) in the APT statements

that refer symbolically to these quantities. All of the geometric types can be nested except TABCYL,

POLCON and RLDSRF. The following examples illustrate some of the applications of nested definitions.

3.21.1. Nested Geometric Definitions Instead of Symbols

SP3 = SPHERE/ CENTER, (POINT/ INTOF L1, L2), TANTO, (PLANE/ $

PTA, PERPTO, (VECTOR/ PTB, PTC))

In this example, the SP3 is defined by its center and a plane to which it is tangent. However, the center of

the sphere is defined by a nested definition of a point that is the intersection of two lines. Similarly, the plane

is defined by a nested definition as passing through a point and perpendicular to a vector that, in turn, is

defined as a nested definition.

3.21.2. Nested Geometric Definitions with Symbols Attached

It is possible to nest the name a geometric type simultaneously. For example:

SP4 = SPHERE/ CENTER, (PT2 = POINT/ INTOF, L1, L2), $

TANTO, PLANEA

The center of the sphere is defined as a nested point definition and has been assigned the symbol PT2.

3.21.3. Nested Geometric Definitions in Non-Geometric Definition Statements

A nested definition can be used in any APT statement where a surface type is referenced.

REFSYS/ (SMATA = MATRIX/ definition)

ZSURF/ (SPLANA = PLANE/ definition)

FROM/ (SPTA = POINT/ definition)

GOTO/ (POINT/ definition)

GOLFT/ (SLA = LINE/ definition)

GOFWD/ (CIRCLE/ definition), PAST, (SLB = LINE/ definition)

3.21.4. Special Case of Nested Definition

Because of the high incident of POINT in other geometric definition formats, a special nesting capability is

provided for points.

In any part of a geometric definition where it would be valid to write:

(POINT/ a, b)

or

(POINT/ a, b, c)

that is, to use a nested point definition without a symbol attached, it would also be valid to write:

(a, b)

or

(a, b, c)

The POINT/ is not required in this case. For example:

C1 = CIRCLE/ P1, P2, (POINT/ 1, 4, 8)

can be written:

C1 = CIRCLE/ P1, P2, (1, 4, 8)

3.21. Nested Definitions

The nested definition permits defining a geometric quantity (except a scalar variable) in the APT statements

that refer symbolically to these quantities. All of the geometric types can be nested except TABCYL,

POLCON and RLDSRF. The following examples illustrate some of the applications of nested definitions.

3.21.1. Nested Geometric Definitions Instead of Symbols

SP3 = SPHERE/ CENTER, (POINT/ INTOF L1, L2), TANTO, (PLANE/ $

PTA, PERPTO, (VECTOR/ PTB, PTC))

In this example, the SP3 is defined by its center and a plane to which it is tangent. However, the center of

the sphere is defined by a nested definition of a point that is the intersection of two lines. Similarly, the plane

is defined by a nested definition as passing through a point and perpendicular to a vector that, in turn, is

defined as a nested definition.

3.21.2. Nested Geometric Definitions with Symbols Attached

It is possible to nest the name a geometric type simultaneously. For example:

SP4 = SPHERE/ CENTER, (PT2 = POINT/ INTOF, L1, L2), $

TANTO, PLANEA

The center of the sphere is defined as a nested point definition and has been assigned the symbol PT2.

3.21.3. Nested Geometric Definitions in Non-Geometric Definition Statements

A nested definition can be used in any APT statement where a surface type is referenced.

REFSYS/ (SMATA = MATRIX/ definition)

ZSURF/ (SPLANA = PLANE/ definition)

FROM/ (SPTA = POINT/ definition)

GOTO/ (POINT/ definition)

GOLFT/ (SLA = LINE/ definition)

GOFWD/ (CIRCLE/ definition), PAST, (SLB = LINE/ definition)

3.21.4. Special Case of Nested Definition

Because of the high incident of POINT in other geometric definition formats, a special nesting capability is

provided for points.

In any part of a geometric definition where it would be valid to write:

(POINT/ a, b)

or

(POINT/ a, b, c)

that is, to use a nested point definition without a symbol attached, it would also be valid to write:

(a, b)

or

(a, b, c)

The POINT/ is not required in this case. For example:

C1 = CIRCLE/ P1, P2, (POINT/ 1, 4, 8)

can be written:

C1 = CIRCLE/ P1, P2, (1, 4, 8)

Chapter 4. Point-To-Point Programming

The APT language provides the capability for explicitly referencing an absolute location to which the cutter

is to be positioned. This concept of moving a cutter from one specific location directly to another specific

location by means of a single straight line movement is called point-to-point programming. The movement

is accomplished with the APT statements FROM, GOTO, and GODLTA. These statements completely

control the motion of the cutter and require no additional information.

4.1. The Initialization (FROM) Statement

The FROM statement specifies a location at which the tip of the cutter is assumed to be positioned. A

FROM statement should be given before any other statement used to define a cutter position; in this regard it

will not result in any movement of the cutter. However, later FROM statements in the part program will

usually result in a movement of the tool. The general format of the FROM statement is as follows:

FROM/ x, y (See Section 4.4)

The FROM point is specified either symbolically or by listing the x, y and z coordinates. The first optional

field is used to specify information concerning the axis of the cutter (either by referencing a symbolic vector

or by giving the individual components). The last optional field is used to specify a feed rate that is to be in

effect until a new value is specified on a later statement.

Note

If the alternate form is used (FROM/ x, y), the optional fields cannot be used.

4.2. The Absolute Movement (GOTO) Statement

The GOTO statement is used to move a tool from its current point of reference to a specified point. The tip

of the cutter will be located at the specified point. The general format of the GOTO statement is as follows:

GOTO/ x, y (See Section 4.4)

The GOTO point is specified either symbolically or by listing the x, y and z coordinates. The two optional

fields have exactly the same meaning as indicated in the FROM statement, and the same rules apply.

4.3. The Incremental Movement (GODLTA) Statement

The GODLTA statement specifies an incremental value that is to be added to each of the coordinates of the

current tool position. This statement, then, defines the amount of movement of the tool in each axis

direction, rather than the specific point to Which the tool is to move (as did the GOTO statement). The

general form of the GODLTA statement is as follows:

The amount to be added to each coordinate can be specified by listing the values (dx, dy, dz) or by

referencing the components of a symbolic vector (done by specifying svector). The third option (specifying

a single scalar value) indicates the tool is to be moved by that amount directly along the tool axis. (A

positive value for delta generates a movement similar to "withdrawing" the tool; a negative value defines a

movement comparable to a plunge operation.) The optional feed rate field has the same meaning as

indicated in the FROM statement,

4.4. Implication of the ZSURF Statement

The ZSURF statement (see Section 3.2) can be used to supply the z coordinate when the alternate form for

the FROM or GOTO statement is used (that is, FROM/ x, y or GOTO/ x, y). The z coordinate is computed

from the plane defined by the ZSURF statement and is zero if no ZSURF statement is specified.

.5. Pattern Definition Statement

A pattern is a set of one or more points. The maximum number of points that can be included in a pattern is

330. Pattern definition statements may be constructed in such a way that a pattern containing the full 330

points cannot be defined. In this case a diagnostic is issued. If this occurs, the desired pattern can always be

defined by breaking the definition statement up into two patterns and adding them together. Even the very

worst case pattern definition statement can always define at least 230 points. It is very unusual to encounter

a pattern definition that cannot define a full 330 points. Patterns may be defined using the formats listed

below. Pattern definitions may not be nested.

4.5.1. Linear (LINEAR) Patterns

A linear pattern is a set of point locations, all of which lie on a straight line.

4.5.1.1. Start Point, End Point and Number of Equally Spaced Points

The start and end points are included in npoints.

SPAT = PATERN/ LINEAR, spoint1, spoint2, npoints

Figure 4.1. LINEAR PATERN 1

PAT1 = PATERN/ LINEAR, PT1, PT2, 9

$$ or

PAT1 = PATERN/ LINEAR, PT1, VEC1, 9

$$ or

PAT1 = PATERN/ LINEAR, PT1, VEC1, INCR, 8, AT, 1

4.5.1.2. Start Point, a Vector, and Number of Equally Spaced Points

The start point is included in npoints. The distance between the points is equal to the vector magnitude. (See

Figure 4.1)

SPAT = PATERN/ LINEAR, spoint, svector, npoints

4.5.1.3. Start Point, a Vector, and an Incremental Spacing

The start point is not included in npoints. The distance between the points is equal to i. (See Figure 4.1)

SPAT = PATERN/ LINEAR, spoint, svector, INCR, npoints, AT, i

4.5.1.4. Starting Point, a Vector and Successive Increments

SPAT = PATERN/ LINEAR, spoint, svector, INCR, i1, i2, ... in

Where i1, i2, ... in, are the successive increments along the vector.

Figure 4.2. LINEAR PATERN 2

PAT2 = PATERN/ LINEAR, PT2, VEC2, INCR, 2, 1, 1, 3

4.5.2. Arc (ARC) Patterns

An ARC pattern is a set of point locations, all of which lie on a single circle. The points may be defined in

the counterclockwise or clockwise direction by specifying the appropriate modifier as follows:

CLW Clockwise

CCLW Counterclockwise

The angles used in arc PATERN definitions are specified in degrees measured counterclockwise from the

positive X axis.

4.5.2.1. A Circle, Start and End Angles and the Number of Points

Where sa and ea specify the starting and ending angles, respectively. The number of points includes the

starting and ending points in the pattern.

Figure 4.3. ARC PATERN

PAT1 = PATERN/ ARC, C1, 70, 230, CCLW, 5

$$ or

PAT2 = PATERN/ ARC, C1, -130, 70, CLW, 5

$$ or

PAT3 = PATERN/ ARC, C1, 70, CLW, INCR, 200, 40, 40, 40

$$ or

PAT4 = PATERN/ ARC, C1, 70, CCLW, INCR, 4, AT, 40

4.5.2.2. A Circle, Starting Angle, and Angular Increments

Where sa is the starting angle and a1, a2, ... an define the angular displacements between successive points

on the circle in the direction specified by CLW or CCLW. (See Figure 4.3)

4.5.2.3. A Circle, Starting Angle, and Number of Increments at an Angular Spacing

Where sa defines the starting angle, and npoints is the number of points (one less than the resultant total

number of points) to be generated at an angular displacement specified by angle in the direction indicated by

CLW or CCLW. (See Figure 4.3)

4.5.3. Parallelogram (PARLEL) Patterns

A parallelogram pattern is a grid of point locations defined by establishing lines parallel to a specified linear

pattern and projecting the points from the original pattern onto those lines. The definition order for the

points in a parallelogram pattern will be governed by the following rules:

1. The first linear pattern in the list defines a "column", the order of which is that of the linear pattern. 2. A parallelogram pattern will consist of at least one other column parallel to the first column. The

points in the second, fourth, and all even-numbered columns will be ordered in the opposite direction

from those in odd-numbered columns.

For example, in Figure 4.4 PAT5 contains points 1-7. Points 8-14 in the next row are ordered in the opposite

direction; points 15-21 are ordered in the same direction as points 1-7, etc. In this example the first column

consists of points 1-7, the second column consists of points 8-14, etc.

Figure 4.4. PARLEL PATERN 1

PAT6 = PATERN/ PARLEL, PAT5, VEC2, INCR, 2.5, 1.5, 1, 2

4.5.3.1. Two Previously Defined Linear Patterns

SPAT = PATERN/ PARLEL, spat1, spat2

spat1 and spat2 must contain a common point as either their first or last point for this definition.

Figure 4.5. PARLEL PATERN 2

PAT4 = PATERN/ PARLEL, PAT1, PAT2

$$ or

PAT4 = PATERN/ PARLEL, PAT1, VEC1, 5

$$ or

PAT4 = PATERN/ PARLEL, PAT1, VEC1, INCR, 5, AT, 4

4.5.3.2. A Linear Pattern, Vector and Number of Columns

SPAT = PATERN/ PARLEL, spat, svect, n

spat1 and spat2 must contain a common point as either their first or last point for this definition. (See

Figure 4.5)

4.5.3.3. A Linear Pattern, Vector, Increment, and Number of Columns

SPAT = PATERN/ PARLEL, spat, svect, INCR, i, AT, n

Where i specifies the increment in units and n specifies the number of times spat is to be repeated at the

specified interval and direction, not including the original pattern. (See Figure 4.5)

4.5.3.4. A Linear Pattern, Vector, and Increments

SPAT = PATERN/ PARLEL, spat, svect, INCR, i1, i2, ... in

Where i1, i2, ... in will be increments (in units) between columns along the direction specified by svect. (See

Figure 4.4)

4.5.4. Random (RANDOM) Patterns

A random pattern is a collection of point locations not necessarily contained on a line or an arc.

4.5.4.1. Previously Defined Points and Patterns

SPAT = PATERN/ RANDOM, spat1, spoint1, spoint2, spat2, spoint3, ...

Figure 4.6. RANDOM PATERN

PAT10 = PATERN/ RANDOM, PAT1, PT8, PT7, PT6, PT5, PT4, PT3$

PT2, PT1

4.6. Point-Point Motion Commands

A cutter movement to a. series of locations may be specified as follows:

GOTO/ spat

where spat is a previously defined pattern.

The resulting output will be a series of GOTO commands specifying movement of the cutter to each point

location defined in the pattern.

The motion between points will be the minimum possible move between points and will be parallel to the

XY plane unless the z-component of the adjacent points differs.

To increase the flexibility of this command the modifiers, OMIT, CONST, INVERS, RETAIN, AVOID, and

THRU are allowed. Their usage is described in Section 4.6.1 through Section 4.6.6.

4.6.1. INVERS Modifier

This modifier specifies that motion should occur to the point locations in a pattern in the reverse order of

their definition. That is, for a pattern containing N points motion would occur to the N, N-1, ..., 1 points.

4.6.2. OMIT Modifier

Points in a pattern may be omitted by specifying OMIT, N1, N2, N3, ..., N, where N1, N2, N3, ..., N

correspond to the points in the pattern according to their output order (affected by INVERS or CONST).

This modifier allows the specification of pattern elements which are not to be included in a specific point-

point motion sequence.

For example, GOTO/ PAT1, OMIT, 2, 3, 7, where PAT1 is illustrated in Figure 4.7, would cause the cutter

to move to points 1, 4, 5, 6 and 8.

Figure 4.7. Sample Pattern 2

4.6.3. RETAIN Modifier

This modifier allows the specification of certain elements of a pattern to be included in a point-point motion

sequence by specifying RETAIN, N1,.N2,N3, ..., N, where N1, N2, N3, ..., N refer to the output order of the

referenced pattern (affected by CONST or INVERS). For example, GOTO/ PAT1, RETAIN, 4, 6, 7, where

PAT1 is illustrated in Figure 4.7, would cause the cutter to move to points 4, 6, and 7.

4.6.4. AVOID Modifier

A vertical displacement (n) from the normal path between points may be maintained While moving the

cutter from Nth point to the (N1 + 1}th point in the referenced pattern, from the Nth point to the (N2 + 1)th

point, by specifying AVOID n, N1, N2, ..., N. The point references refer to the output order of the pattern

(affected by INVERS and CONST). For example, GOTO/ PAT1, AVOID, 3, 2, 4, would result in the cutter

movement illustrated in Figure68h. (This figure is missing in the original manual -- SED)

4.6.5. THRU Modifier

This modifier is used to specify a range of points for the OMIT, RETAIN, and AVOID modifiers.

For example, GOTO/ PAT1, RETAIN, 2, THRU, 6 would cause the cutter to move to the 2, 3, 4, 5, and 6

points of PAT1.

When using the THRU modifier, the range does not have to be specified in the order of movement. For

example, GOTO/ PAT1, RETAIN, 6, THRU, 2 would also cause the cutter to move to the 2, 3, 4, 5 and 6

points of PAT1.

4.6.6. CONST Modifier

The CONST modifier allows the programmer to specify the point numbers for the OMIT, RETAIN, or

AVOID modifiers according to the definition order instead of according to the output order.

The example in Figure 4.8 illustrates the use of this modifier. GOTO/ PAT1, INVERS, RETAIN, 2, THRU,

4, 6, 8 and GOTO/ PAT1, INVERS, CONST, RETAIN, 11, THRU, 9, 7, 5 will both cause the cutter to

move to the points numbered 11, 10, 9, 7, 5.

Figure 4.8. Sample Pattern 1

4.7. Point-Point Programming Considerations

INVERS, AVOID, and either RETAIN or OMIT may be combined in a single GOTO/ SPAT statement. The

following considerations further explain the effect of combining these modifiers.

4.7.1. Ordering of the Operations Performed on a Pattern

The order of operations (OMIT/RETAIN or AVOID) must be specified in the output sequence of the points.

For example, an AVOID is to occur between the fourth and fifth points of a pattern, the seventh point is to

be omitted, and another AVOID is to occur between the tenth and eleventh points. The modifiers may be

stated as follows:

GOTO/ SPAT, AVOID, 3, 4, OMIT, 7, AVOID, 3, 10

It is permissible to overlap a range of operations of one type with a single operation of another type. For

example, if the fourth through the tenth points and the twelfth through the fifteenth points were to be

retained and AVOID was to occur between the thirteenth and fourteenth points, the following statement

would be permissible:

GOTO/ SPAT, AVOID, 2, 13, RETAIN, 4, THRU, $

10, 12, THRU, 15

If an AVOID was to occur between the sixth and seventh points and the eighth and ninth points, however,

the following statement would be used:

GOTO/ SPAT, RETAIN, 4, THRU, 10, AVOID, 2. 5, $

6, 8, RETAIN, 12, THRU, 15

Chapter 5. Programming a Tool Path

The APT language can be used to direct a cutter along a predetermined path. This chapter introduces the

fundamental language concepts required to accomplish this task. Expansion of some of the more complex

concepts and definition of additional language features are made in Chapter 14. Guidelines and suggested

procedures are stated only in general, since experience is the only way good techniques and reliable methods

are formulated.

Experience in the numerical control industry has shown that the geometric problems involved in APT must

be solved using iterative techniques. Other mathematical approaches are specific in nature and fail to

achieve the necessary degree of generality. Even the more desirable iterative approach used in this processor

may fail in unforeseeable situations. In such circumstances minor part program modifications can be made

to achieve the required results.

5.1. Introduction

Even a brief description of how the tool motion is controlled depends upon an understanding of several

concepts. The individual notations are discussed in Section 5.1.1 through Section 5.1.3, and there

relationships are discussed in Section 5.1.3.4.

5.1.1. Description of Cutter

The cutter shape must be defined before any cutter motion can be specified. The format of the statement that

defines the cutter is:

CUTTER/ d, r, e, f, , , h

Note

The parameters are as shown in Figure 5.1. The parameters must all be positive

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