1 Chapter 7 Geometric Transformation 3D transformations 3D affine transforms including translation, rotation, scaling, shearing, and reflections

1Zhang & Liang, Computer Graphics Using Java 2D and 3D (c) 2007 Pearson Education, Inc. All rights reserved.

Chapter 7 Geometric Transformation

3D transformations 3D affine transforms including translation,

rotation, scaling, shearing, and reflections Transformation matrices Applying transforms in scene graphs Composite transformations Using transforms in constructing geometries


Transformations in 3D

Transformations are used to– define and manipulate objects– project from 3D to 2D

Different levels of abstraction– Matrix4d– Transform3D– TransformGroup


Types of Transformations

Affine transform maps lines to lines – preserves parallelism

Rigid motions (aka isometries, Euclidean motions) preserve shape

There are the same types of transformations in 3D as in 2D but they can be more complicated


Representing 3D Transformations

3D affine transformation

In homogeneous coordinates


Matrix Classes

Low-level representation for transformations Matrix classes for 3x3 and 4x4 matrices for both

float and double typeMatrix3fMatrix3dMatrix4fMatrix4d

GMatrix can be used for matrices of arbitrary size, not necessarily squareGMatrix


Matrix Operations

Update the current matrix (similar to compound assignment)void add(Matrix4d m1)void sub(Matrix4d m1)void mul(Matrix4d m1)

Combine two MatrixObjects and put result into current Matrixvoid add(Matrix4d m1, Matrix4d m2)void sub(Matrix4d m1, Matrix4d m2)void mul(Matrix4d m1, Matrix4d m2)


Matrix Operations

Matrix times its inverse gives identity matrixvoid invert()void invert(Matrix4d m1)

Exchange rows and columnsvoid transpose()

Multiply all elements by the same valuevoid mul(double scalar)

Determinant as a scalar value associated with a matrix (sort of like the magnitude of a vector)double determinant()


Transform3D

A higher-level representation for transformations

Has constructors to create from both Matrix objects and from vectors

Also has set and get methods for the internal 4x4 matrix

Can also set a particular type of transformation


Translation in 3D Translation can be specified by a vector

which becomes the fourth column of the transformation matrixvoid set(Vector3d trans)void set(Vector3f trans)void setTranslation(Vector3d trans)void setTranslation(Vector3f trans)


Scaling in 3D

Can scale the entire object uniformly by specifying a single scale factorvoid set(double scale)void setScale(double scale)

Can scale the object non-uniformly by specifying a vector with a scale factor for each directionvoid setScale(Vector3d scales)


Reflection in 3D

In 3D, reflection is performed into a plane Reflection of plane through origin with

normal vector u can be expressed as– vector equation

Matrix representation:– inverse is same matrix


Shear in 3D

In 3D, shear along a plane Can affect a single coordinate

x' = x + shx z

or two coordinatesx' = x + shx z

y' = y + shy z


3D Rotation In 3D, can rotate about an

arbitrary line– hard to come up with a matrix for an

arbitrary rotation AxisAngle4d allows you to

specify the line to rotate about– AxisAngle4d( x, y, z, q)

represents rotation by angle about the vector (x, y, z)

Set a general rotation in Transform3D using AxisAngle4dvoid set(AxisAngle4d r)


Quaternions

Quaternions provide an extension of complex numbers– instead of two values, a quaternion has 4

An arbitrary quaternion has the formp = A + b i + c j + d k

i, j and k satisfyi2 = j2 = k2 = ijk = -1

Tuple operations plus conjugate and inverse


Quaternions for Rotation

A point represented as a pure quaternion– no real partp = x i + y j + z k

A rotation defined by quaternion operations

– θ is the angle of rotation and u defines the axis of rotation through the origin


Euler Angles

Another way to think about rotations in 3DUse three rotations about the coordinate

axes– elevation, azimuth, tilt– roll, pitch, yaw– precession, nutation, spin – heading, altitude, bank

Transform3D methodvoid setEuler(Vector3d eulerAngles)


Quaternion to Euler Angle

No Transform3D method for retrieving Euler angles

Compute them from the quaternionpublic static Vector3d quatToEuler(Quat4d q1) { double sqw = q1.w*q1.w; double sqx = q1.x*q1.x; double sqy = q1.y*q1.y; double sqz = q1.z*q1.z; double heading = Math.atan2(2.0 * (q1.x*q1.y + q1.z*q1.w), (sqx - sqy - sqz + sqw)); double bank = Math.atan2(2.0 * (q1.y*q1.z + q1.x*q1.w), (-sqx - sqy + sqz + sqw)); double attitude = Math.asin(-2.0 * (q1.x*q1.z - q1.y*q1.w)); return new Vector3d(bank, attitude, heading);}

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TestTransform.java

Use this to explore different transformations– Enter a transformation matrix and see what it

does– Enter data for particular kind of transformation

and see what it does

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Transform3D Node

Within a SceneGraph, a TransformGroup node implements a transformation that is applied to all of its children

This allows us to build hierarchical scenes


Scene Graph for Axes class

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Rotation.java

Create ColorCubes Use a series of rotation transformations to

position the cubes uniformly in a ring around an axis


Composite Transforms

Composition of T1 and T2 is(T2T1)(P) = (T2 ( T1 (P) ) )

Order matters– The first transformation to be applied is the

second in the list

T2T1 != T1T2

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A Common Pattern

Arbitrary transformations are hard to write matrices for– Rotation about an arbitrary axis– Reflection into an arbitrary plane

Solution: T-1RT– Translate the scene so that the desired operation

is trivial to write down – Do the desired transfromation– Translate back after the transformation

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Example 1

Rotation about an arbitrary axis– / 3 rotation about line through (1,1,0) and

(1,2,1) Translate by (-1, -1, 0) so first point goes through

origin Use a quaternion to specify the rotation Translate by (1, 1, 0) to get back to original position

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Example 2: Mirror.java

Reflection into an arbitrary planea x + b y + c z = 0

Solution– Rotate to map the plane into the x-y plane

d = sqrt( a2 + b2)cos() = c / sqrt( a2 + b2 + c2)q = cos(/2) + (b/d i - a/d j) sin (/2)

– Reflection into x-y plane (z -> -z)– Do inverse rotation to get back to original


Transform3D methods

Transform Point3D objectsvoid transform(Point3d p)void transform(Point3d p, Point3d pOut)void transform(Point3f p)void transform(Point3f p, Point3f pOut)

Transform Vectorsvoid transform(Vector3d v)void transform(Vector3d v, Vector3d vOut)void transform(Vector3f v)void transform(Vector3f v, Vector3f vOut)void transform(Vector4d v)void transform(Vector4d v, Vector4d vOut)void transform(Vector4f v)void transform(Vector4f v, Vector4f vOut)


Shape Construction by Extrusion

Extend a 2D shape to 3D– Get a PathIterator for the shape– Construct quadrilateral (or triangular) polygon

mesh between successive points along the path– see extrudeShape.java for method that does this

Java3D provides a class to do this automatically to create Font3D objects– See Hello3D example from chapter 5

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PathIterator

An interface that allows Shape objects to return an iterator to their sub-parts

Defines constants to represent the different types of path segments

Methodsint currentSegment( double [] coordinates)

boolean isDone()

void next()

int getWindingRule()


Shape Construction by Rotation

Many shapes can be created by rotating a curve around an axis– create a polygon mesh between points at

successive angles Example: constructing a torus

– See Torus.java, TestTorus.java


Transformation and Shared Branch

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1 Chapter 7 Geometric Transformation 3D transformations 3D affine transforms including translation, rotation, scaling, shearing, and reflections