LOCI OF POINTS AND LOCI OF POINTS AND GENERATED CURVESGENERATED CURVES

Loci of PointsLoci of Points

The path traced out by a point when it The path traced out by a point when it moves in space, under given conditions or moves in space, under given conditions or in accordance with a definite law, is known in accordance with a definite law, is known as locus of that point (loci is the plural of as locus of that point (loci is the plural of locus)locus)

The path of a point which moves The path of a point which moves according to mathematically defined according to mathematically defined conditions is known as its Locus.conditions is known as its Locus.

LOCUSLOCUS

For example, a point P moving in a plane, For example, a point P moving in a plane, so that it is always at a constant distance so that it is always at a constant distance from another fixed point O traces out a from another fixed point O traces out a circle as its locus. circle as its locus.

Many important geometrical curves Many important geometrical curves (ellipse, parabola, hyperbola, cycloidal (ellipse, parabola, hyperbola, cycloidal curves) may be considered as Loci e.g, curves) may be considered as Loci e.g, conic curves, helices, and screw threads, conic curves, helices, and screw threads, involutes and spiral curves.involutes and spiral curves.

Locus as a circleLocus as a circle

The locus of a point P is a circle when it The locus of a point P is a circle when it moves in a plane in such a way that its moves in a plane in such a way that its distance from a fixed point O, always distance from a fixed point O, always remain constant. The fixed point O is remain constant. The fixed point O is called the centre and the constant called the centre and the constant distance OP is called the radius.distance OP is called the radius.

P

O

Locus as a straight lineLocus as a straight line The locus of a point P is a straight line when it moves in a plane in The locus of a point P is a straight line when it moves in a plane in

such a way that its distance from a fixed line AB is always a such a way that its distance from a fixed line AB is always a constant. constant.

If the fixed line AB is an arc of a circle, then the locus will be If the fixed line AB is an arc of a circle, then the locus will be another arc drawn through the point P and having the same center another arc drawn through the point P and having the same center as of arc AB. as of arc AB.

A B

Locus of P

Locus of point on a mechanismLocus of point on a mechanism

The locus of a point in a mechanism is the path which is The locus of a point in a mechanism is the path which is traced by the point when the mechanism moves through traced by the point when the mechanism moves through a complete cycle of operation. a complete cycle of operation.

The method of drawing the locus of a particular point in a The method of drawing the locus of a particular point in a mechanism is to construct the mechanism in several mechanism is to construct the mechanism in several positions. positions.

The point is plotted for each position and its locus is The point is plotted for each position and its locus is obtained by drawing a smooth curve through these obtained by drawing a smooth curve through these plotted points.plotted points.

The mechanism in successive positions may be drawn The mechanism in successive positions may be drawn with drawing instruments geometrically or with a paper with drawing instruments geometrically or with a paper trammel. The use of computer aided drafting renders the trammel. The use of computer aided drafting renders the procedure very handy and fast. procedure very handy and fast.

Plane and space curvesPlane and space curves

Curve or curved lineCurve or curved line A line which is generated by a point that moves in a A line which is generated by a point that moves in a

constantly changing direction is called curve or curved constantly changing direction is called curve or curved line.line.

The exact nature of each curve or curved line is The exact nature of each curve or curved line is determined by the motion of its generated point. The determined by the motion of its generated point. The following are the two general classes of curves or curved following are the two general classes of curves or curved lines :-lines :- Plane curves or single-curved linesPlane curves or single-curved lines

A line which is generated by a point that moves in a constantly A line which is generated by a point that moves in a constantly changing direction in the same plane is called plane curve or single changing direction in the same plane is called plane curve or single curved line.curved line.

Space curves or Double curved linesSpace curves or Double curved linesa line which is generated by a point that moves in a a line which is generated by a point that moves in a

constantly changing direction in the space is called space curve constantly changing direction in the space is called space curve or double curved line.or double curved line.

Types of plane and space curvesTypes of plane and space curves

The following are the important types of plane and space curves The following are the important types of plane and space curves used in engineering practice :-used in engineering practice :-

1.1. Plane curves or single curved lines:Plane curves or single curved lines:(a)(a) Roulettes or cycloidal curvesRoulettes or cycloidal curves

(i)(i) CycloidCycloid Trochoids (superior and inferior)Trochoids (superior and inferior)(ii)(ii) Epicycloid Epicycloid

Epitrochoids (superior and inferior)Epitrochoids (superior and inferior)(iii)(iii) HypocycloidHypocycloid

Hypotrochoid (superior and inferior)Hypotrochoid (superior and inferior)(b)(b) InvolutesInvolutes(c)(c) SpiralsSpirals

(i)(i) Archemedian spiralArchemedian spiral(ii)(ii) Logarithmic spiralLogarithmic spiral

2.2. Space curves or double curved lines:Space curves or double curved lines:(a)(a) HelixHelix

(i) Cylindrical Helix(i) Cylindrical Helix(ii) Conical helix(ii) Conical helix

1. PLANE CURVES OR SINGLE CURVED LINES1. PLANE CURVES OR SINGLE CURVED LINES (a) (a) CYCLOIDAL AND SPIRAL CURVESCYCLOIDAL AND SPIRAL CURVES

ROULETTES or CYCLOIDAL CURVESROULETTES or CYCLOIDAL CURVESThose curves which are generated by a Those curves which are generated by a

fixed point on a rolling curve that rolls fixed point on a rolling curve that rolls without slipping along fixed base curve. without slipping along fixed base curve.

The rolling curve is called generating The rolling curve is called generating curve and the fixed curve is called the curve and the fixed curve is called the directing curve. directing curve.

APPLICATION OF ROULETTESAPPLICATION OF ROULETTES

CYCLOIDCYCLOID

The curve is the locus of a point on the The curve is the locus of a point on the circumference of a circle which rolls, circumference of a circle which rolls, without slipping, along a fixed straight line.without slipping, along a fixed straight line.

ENGINEERING APPLICATION OF ENGINEERING APPLICATION OF CYCLOIDCYCLOID

The cycloid curve was formerly used more The cycloid curve was formerly used more extensively in the design of gear tooth extensively in the design of gear tooth profile, but modern production methods profile, but modern production methods tend to limit its applications to small gears tend to limit its applications to small gears used in instruments, for which epicycloids used in instruments, for which epicycloids or hypocycloid curves are generally used. or hypocycloid curves are generally used.

PROBLEMPROBLEM

Draw a cycloid, given the diameter of a Draw a cycloid, given the diameter of a generating circle as 50 mm. also draw a generating circle as 50 mm. also draw a tangent and normal at any given point T tangent and normal at any given point T on the curve.on the curve.

Solution - CycloidSolution - Cycloid

Solution - StepsSolution - Steps

With center CWith center Coo draw the rolling circle of 50 mm. draw a draw the rolling circle of 50 mm. draw a straight line, the path along which it is to roll, tangent to straight line, the path along which it is to roll, tangent to the circle.the circle.

Fix the initial position of point which is to trace the Fix the initial position of point which is to trace the required locus while the rolling circle make some required locus while the rolling circle make some revolution along the base line. Let it be Prevolution along the base line. Let it be Po. o.

Mark a length PMark a length Po o PPo o equal to the circumference of the equal to the circumference of the rolling circle, along the base line, and divide it into a rolling circle, along the base line, and divide it into a number of equal parts, 12 here. Divide the circumference number of equal parts, 12 here. Divide the circumference of the rolling circle also into the same number of equal of the rolling circle also into the same number of equal parts. parts.

Solution - StepsSolution - Steps Through division points on the rolling circle, draw lines Through division points on the rolling circle, draw lines

parallel to fixed line and at the points on the fixed line parallel to fixed line and at the points on the fixed line erect perpendiculars to cut the horizontal center line of erect perpendiculars to cut the horizontal center line of the rolling circle at points Cthe rolling circle at points C11, C, C22, C, C33 etc. etc.

As the circle rolls through 1/12As the circle rolls through 1/12thth of a complete revolution, of a complete revolution, the center Co will move to the position C1 and the point the center Co will move to the position C1 and the point P will move from initial position Po to P1 and so on. P will move from initial position Po to P1 and so on. Therefore, the points Po, P2, P3 etc. are plotted by the Therefore, the points Po, P2, P3 etc. are plotted by the intersection of lines drawn division points 1, 2, 3 etc on intersection of lines drawn division points 1, 2, 3 etc on the circle and the corresponding circle arcs drawn with the circle and the corresponding circle arcs drawn with centers C1, C2 etc, as illustrated for P4 and P5.centers C1, C2 etc, as illustrated for P4 and P5.

A smooth curve joining all the 12 points plotted thus, A smooth curve joining all the 12 points plotted thus, gives the required cycloid. gives the required cycloid.

Solution - StepsSolution - Steps

Tangent and normal at a point on the cycloidTangent and normal at a point on the cycloidDraw the rolling circle in such a position Draw the rolling circle in such a position

that It passes through T, by chain line. The that It passes through T, by chain line. The normal is given by the line TN, where N is normal is given by the line TN, where N is the point of contact between the rolling the point of contact between the rolling circle, and the fixed line. The tangent T1, circle, and the fixed line. The tangent T1, T2 is perpendicular to TN at T.T2 is perpendicular to TN at T.

TrochoidsTrochoids The curve generated by a point within or outside the The curve generated by a point within or outside the

circle which rolls along a straight line is called trochoid.circle which rolls along a straight line is called trochoid. When a circle rolls, without slipping along a fixed straight When a circle rolls, without slipping along a fixed straight

line, the locus of the fixed point P not lying on the rolling line, the locus of the fixed point P not lying on the rolling circle is a trochoid.circle is a trochoid.

When the point P which traces the locus is outside the When the point P which traces the locus is outside the rolling circle, the locus produced is rolling circle, the locus produced is superior trochoidsuperior trochoid..

When the point P is inside the rolling circle the locus is When the point P is inside the rolling circle the locus is inferior trochoidinferior trochoid..

The construction of both trochoids is very similar to that The construction of both trochoids is very similar to that used for cycloid.used for cycloid.

ProblemProblem

Draw trochoids, given the diameter of the Draw trochoids, given the diameter of the rolling circle as 40 mm and the fixed point rolling circle as 40 mm and the fixed point P, tracing the locus, is 8 mm away from P, tracing the locus, is 8 mm away from the rolling circle.the rolling circle.

Solution - Superior TrochoidSolution - Superior Trochoid

Solution - Inferior TrochoidSolution - Inferior Trochoid

Solution Solution

The construction of both trochoids is very The construction of both trochoids is very similar to that used for cycloid. It should be similar to that used for cycloid. It should be noted however, that in each case the noted however, that in each case the circumference of the rolling circle is laid circumference of the rolling circle is laid out along the fixed line and divided into 12 out along the fixed line and divided into 12 equal parts, and the circle through the equal parts, and the circle through the given point P is divided into 12 equal given point P is divided into 12 equal parts, not the reverse. parts, not the reverse.

EpicycloidEpicycloid

The curve generated by a point on the The curve generated by a point on the circumference of a rolling circle which rolls circumference of a rolling circle which rolls outside the directing circle is called epicycloid.outside the directing circle is called epicycloid.

When a circle rolls, without slipping, around the When a circle rolls, without slipping, around the outside of a fixed circle, the locus of a point on outside of a fixed circle, the locus of a point on the circumference of the rolling circle is called the circumference of the rolling circle is called the epicycloid.the epicycloid.

The rolling circle is called The rolling circle is called generating circlegenerating circle and and the fixed circle is called the the fixed circle is called the directing circledirecting circle..

ProblemProblem

Draw an epicycloid, given the radii of Draw an epicycloid, given the radii of rolling and directing circles as r = 30 mm rolling and directing circles as r = 30 mm and R = 120 mm, respectively. Also draw and R = 120 mm, respectively. Also draw a normal and a tangent at any point Q on a normal and a tangent at any point Q on the curve. the curve.

Solution - EpicycloidSolution - Epicycloid

1. PLANE CURVES OR SINGLE CURVED LINES1. PLANE CURVES OR SINGLE CURVED LINES (b) Involute(b) Involute

A curve traced out by an end of a piece of A curve traced out by an end of a piece of string when unwound from a circle or a string when unwound from a circle or a polygon is called involute.polygon is called involute.

When a straight line rolls, without slipping, When a straight line rolls, without slipping, on a curve, the locus of any point on the on a curve, the locus of any point on the straight line is an involute to the curve. straight line is an involute to the curve.

The involute to a circle is the locus of the The involute to a circle is the locus of the end of a taut string as it is unwound from end of a taut string as it is unwound from the surface of a cylinder or base circle.the surface of a cylinder or base circle.

Engineering application of involuteEngineering application of involute

Involute of a circle is used as the profile of gear teeth. Involute of a circle is used as the profile of gear teeth. Cams are often designed to the involute shape because Cams are often designed to the involute shape because it ensures rolling contact between the roller and the it ensures rolling contact between the roller and the follower at constant speed.follower at constant speed.

The involute of a circle can be drawn by drawing The involute of a circle can be drawn by drawing tangents at various points on the circumference of the tangents at various points on the circumference of the circle and making the various points at corresponding circle and making the various points at corresponding distances along their Tangents. While the involute of any distances along their Tangents. While the involute of any polygon can be drawn by extending its sides, keeping polygon can be drawn by extending its sides, keeping the corners of polygon as successive sides of the the corners of polygon as successive sides of the polygon thereby terminating on the extended sidespolygon thereby terminating on the extended sides

ProblemProblem

Draw an involute to a circle of 50 mm. Also Draw an involute to a circle of 50 mm. Also draw a tangent and normal to it, at any draw a tangent and normal to it, at any given point on it. given point on it.

Solution – Involute to a circleSolution – Involute to a circle

ProblemProblem

Draw the involute of a circular arc which Draw the involute of a circular arc which subtends an angle (90 degrees here) at subtends an angle (90 degrees here) at the center of the circle of 120 mm. the center of the circle of 120 mm.

Involute to a circular arcInvolute to a circular arc

ProblemProblem

Draw an involute to an equilateral triangle Draw an involute to an equilateral triangle of 20 mm side. of 20 mm side.

Involute of a triangleInvolute of a triangle

1. PLANE CURVES OR SINGLE CURVED LINES1. PLANE CURVES OR SINGLE CURVED LINES (c) Spirals(c) Spirals

A curve generated by a point moving continuously in one A curve generated by a point moving continuously in one direction along a rotating line is called spiral.direction along a rotating line is called spiral.

The point or the end about which the line rotates is The point or the end about which the line rotates is called pole.called pole.

The line joining any point on the spiral curve with the The line joining any point on the spiral curve with the pole is called the radius vector and the angle between pole is called the radius vector and the angle between this and the line in its initial position is called the vector this and the line in its initial position is called the vector angle. angle.

When line completes one revolution, the moving point is When line completes one revolution, the moving point is said to have traced out one revolution. A spiral may take said to have traced out one revolution. A spiral may take any number of revolutions before reaching the pole, but any number of revolutions before reaching the pole, but there will be as many convolutions as the number of there will be as many convolutions as the number of revolutions.revolutions.

Archimedean SpiralArchimedean Spiral

The curve traced out by a point moving with The curve traced out by a point moving with uniform velocity along a line which is also uniform velocity along a line which is also rotating with uniform velocity is called rotating with uniform velocity is called Archimedean spiral. Archimedean spiral.

It is the locus of a point P which moves at a It is the locus of a point P which moves at a steady rate along a line, while the line rotates at steady rate along a line, while the line rotates at uniform speed about center, O , such that for uniform speed about center, O , such that for each angular displacement of the line, the linear each angular displacement of the line, the linear displacement of the point is constant. displacement of the point is constant.

Engineering applications of Engineering applications of Archimedean Spirals Archimedean Spirals

They are used in the construction of cams, They are used in the construction of cams, threads of scroll chucks and in some other threads of scroll chucks and in some other simple devices. simple devices.

ProblemProblem

Construct an Archimedean spiral of two Construct an Archimedean spiral of two convolutions, given the greatest and the convolutions, given the greatest and the shortest radii as 84 mm and 12 mm, shortest radii as 84 mm and 12 mm, respectively.respectively.

Archimedean Spiral ( Two Convolutions )Archimedean Spiral ( Two Convolutions )

ProblemProblem

Construct an Archimedean spiral of one Construct an Archimedean spiral of one convolution , given the radial movement of convolution , given the radial movement of the point P during one convolution as the point P during one convolution as 60mm and the initial position of P as pole 60mm and the initial position of P as pole O. O.