CSC418 - Notes
Topic 2: Basic Geometric Modeling

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Key Concepts & Readings

Coordinate-Free Geometry
2D Transformations (Shirley: p. 85-96)
Transformation Matrices (Shirley: p. 85)
Composing 2D Transformations (Shirley: p. 91-96)
Scene Graphs (Shirley: p. 224-225. Hill: p. 258-262. Foley & van Dam: p. 280, 867)
Homogeneous Coordinates
Midpoint Algorithm (Shirley: p. 55)
Opengl (The Red Book:, c. 3)

OpenGL

Transformation Matrices

glMatrixMode(GL_MODELVIEW)	selects the model-view matrix for subsequent matrix operations
glTranslate(x,y,z)	constructs translation matrix and multiply the currently selected matrix with the translation matrix
glRotate(th_rad,vx,vy,vz)	constructs rotation matrix and multiply the currently selected matrix with the rotation matrix
glScale(x,y,z)	constructs scaling matrix and multiply the currently selected matrix with the scaling matrix
glLoadIdentity()	sets the currenlty selected matrix to identity
glPushMatrix()	saves the currently selected matrix to the matrix stack
glPopMatrix()	sets the currently selected matrix equal to the top entry of the matrix stack and pops the stack
glLoadMatrix(... m)	sets the currently selected matrix equal to the t
glMultMatrix(... m)	replaces the current matrix with m * current matrix

Matrix M can be setup as follows:

  glMatrixMode( GL_MODELVIEW );
  glLoadIdentity();
  glTranslatef(2.0, 1.0, 0.0);
  glRotatef( -3.14/2.0, 0.0, 0.0, 1.0); 
  glScalef(2.0, 2.0, 2.0);
  ...

Which produces the matrix M = trans(2,1,0) rot(z,-90) scale(2,2,2) ....
Another way of loading M is to use:

  glMatrixMode( GL_MODELVIEW );
  glLoadMatrixf( A );
  ...

Mere, M = A
Yet another way of constructing a model-view matrix is:

  glMatrixMode(GL_MODELVIEW);
  glLoadIdentity();
  glMultMatrix(A);
  glMultMatrix(B);

Here, M = B A I

Using Transformations

The following opengl code draws four instances of a square using geometric transformations. DrawSquare() draws a unit square centered at (0,0), i.e, a square with unit area.

  ...   
  glMatrixMode(GL_MODELVIEW);     /* select the model-view matrix for subsequent matrix operations */ 
  glLoadIdentity(); /* make the model-view matrix identity */
  DrawSquare(); /*  draw the square 1 */
  glTranslatef(5.0, 0.0, 0.0); /* translate along z-axis by -5 units*/
  DrawSquare; /*  draw the square 2 */
  glScalef (2.0, 5.0, 1.0); /* scale. notice that scaling can be different along each dimension */
  DrawSquare(); /* draw square 3. notice how it is affected by previous transformations (translation & rotation), so one can combine transformations. */ 
  glLoadIdentity (); /* we need to clear the matrix; otherwise the translation will also apply to the second square */  
  glScalef (3.0, 2.0, 1.0); /* scale. notice that scaling is different along each dimension */
  DrawSquare(); /* draw the square 4 */
  ...

 
 Four  instances of a square are drawn using the same primitive "square drawing" function by using transformations.

Scene Graph

The following code generates the above figure.

  ...
  glMatrixMode(GL_MODELVIEW); /* select the model-view matrix for subsequent matrix operations */ 
  glLoadIdentity(); /* model-view matrix = identity */
  glMultMatrix(m); /* model-view matrix = m * model-view matrix */ 
  DrawLink(); /* draws Link 1 */
  glPush(); /* saves the model-view matrix */
  glTranslate(x1,y1,z1); /* model-view = translation * model_view */
  DrawLink(); /* draws Link 2 */
  glRotation(th_rad,vx1,vy1,vz1);    /* model-view = rotation * model-view */
  glTranslation(x2,y2,z2); /* model-view = translation * model_view */
  DrawLink(); /* draws Link 4 */ 
  glPop(); /* model-view matrix = identity */
  glPush(); /* again save model-view matrix*/
  glRotation(th_rad,vx2,vy2,vz2);    /* model-view = rotation * model-view */
  glTranslation(x3,y3,z3); /* model-view = translation * model_view */
  DrawLink(); /* draws Link 3 */
  glPop(); /* model-view matrix = identity */ 
  ...

Notes

Coordinate-Free Geometry

pdf file

2D Geometric Transforms

Also see 3D Geometric Transforms

Translation, rotation, scaling, and shearing are all examples of geometric transformations. Multiple copies of an object can be created by drawing the same object using a series of different transformation matrices.

(image courtesy of Robert Lansdale)

Affine Transformation

A transformation that preserves the parallelism of lines, but not necessarily angles or lengths.

Derivations

Suppose we transform an object (by translation, scale, rotation, shear) then a point p(x,y) of the object will be moved another point p'(x',y') of the transformed object. We want to derive the relationship between p' and p.

translation	trans(a,b): x'=x+a y'=y+b
scaling	scale(a,b): x'=a x y'=b y
shear	shearx(a): x'=x+a y y'=y sheary(a): x'=x y'=y+a x
rotation	Imagine z-axis coming out of the paper. We define counter clock-wise rotation about z-axis as positive rotation and clock-wise rotation about z-axis as negative, i.e., we are following the right hand convention. If you align your right thumb with the positive z-axis (coming out of the page), the direction of the curl of your fingers is positive rotation. rot(zaxis,angle_rad): x'=x cos(a) - y sin(a) y'=x sin(a) + y cos(a)

Homogeneous Coordinates

Represent each cartesian point (x,y) by 3 coordinates, (hx,hy,hw).
Setting w=0 can be thought of as a point in infinity, or simply, a direction
Allows us to describe a translation transformation as a transformation matrix (see Transformation Matrices)

Transformation Matrices

The transformations defined above can be rewritten using 3x3 matrices and homogeneous coordinates. The general form of the matrix is:

The expressions derived earlier correspond to the following transformation matrices.

Translation Matrix
Scaling Matrix
Rotation Matrix

Composition and Inversion of Transformations

A series of transformations can be accumulated into a single transformation matrix. Suppose our object is defined as follows:

Suppose we wish to place the object in our scene like this:

This can be accomplished in several ways, one of which is:

Substituting (1) in (2) gives:

Comments on Matrix Composition

compositions of transformations are non-commutative:
trans(2,3)rot(z,-90)!= rot(z,-90)trans(2,3)
transformations matrices can be multiplied together to achieve a single composite transformation matrix
the placement of a transformation determines the coordinate system in which it takes place. For example, in making an additional transformation to our model as follows, we can choose to express the translation in either object or world coordinates.
As another example, we could rethink our initial example as having a trans(2,3) executed, followed by a rot(z,-90) in object coordinates. Thus, if we specify a sequence of transformations always in terms of a fixed (world) coordinate system, they should be ordered from right-to-left. On the other hand, if we think of all transformations in terms of a local (object) coordinate system, they should be ordered from left-to-right.
Transformations are invertible, either by inverting the matrix or by undoing the transformations. i.e., trans(a,b) trans(-a,-b) = I
Transformations can be thought of as a change in coordinate system.
The transformations necessary to convert a point from CS 2 to CS 1 are given by the transformations necessary to align the CS 1 frame with the CS 2 frame.

Hiearchical Models

Scene Graphs OR Transformation Hierarchies

Consider building the following model of a hand with one finger:

This can be constructed using a transformation hierarchy. In the following scene graph, circles represent transformations and squares represent geometry. The pseudocode on the left can be used to draw the scene.

f1: trans(d_hand,0) rot(z,th1)
f2: trans(d1,0) rot(z,th2)
f3: trans(d2,0) rot(z,th3)

M=M*Thand
draw_hand
M=M*Tf1;
draw f1;
M=M*Tf2;
draw f2;
M=M*Tf3;
draw f3

Now consider drawing a hand with three identical fingers. We can create a more complex scene graph which uses multiple instances of a finger scene graph, as shown below. Because each of the fingers is defined relative to the hand coordinate system, a way is needed to restore the hand coordinate system before beginning to draw each finger. This is done through the pushMatrix() and popMatrix() function calls.

M=M*Thand
draw_hand
pushMatrix()
M=M*Tf1a
draw_finger()
popMatrix()
pushMatrix()
M=M*Tf1b
draw_finger()
popMatrix()
pushMatrix()
M=M*Tf1c
draw_finger()
popMatrix()

draw_finger() {
draw f1
M=M*Tf2
draw f2
M=M*Tf3
draw f3
}

Many graphics systems maintain a stack for the current transformation matrix:

Exercises

Exercise 1

Assuming 2D coordinates (3x3 homogeneous matrices), show that rotations and translations do not commute.
Derive a 3x3 homogeneous matrix that scales an object by a factor S, along the line y = m x.

Exercise 2

An affine transformation can be completely specified by its action on three points. What is the matrix for the transformation that maps the points A(0,0) B(1,0) and C(1,0) to the points A'(1,2) B'(1,1) and C'(2,1) respectively?

Hint: 2D rotation, scaling, shear and translation are all affine transformations, of the form

p' = Mp, where p=[x,y,1]^T and M is a 3x3 matrix of the form:

[	a	b	c	]
[	d	e	f	]
[	0	0	1	]

Exercise 3

Assuming two-dimensional transformations, show that the following pairs of operations do or do not commute:

Two rotations commute.
A rotation and a uniform scaling (Sx=Sy) commute.
A rotation and a non-uniform scaling do not commute.

Exercise 4

Reflections: Assume a two-dimensional space.

Give the homogeneous (3 x 3) matrix representing a reflection about the x axis.
Give the matrix for a reflection about a line which passes through the origin and makes an angle a with the x-axis (Hint: how do we rotate around non-origin points?).
Given an n by n raster of pixels such that pixel a[x][y] is displayed at (x,y), write a code fragment to reflect the pixels about the line x=y.
Reflect the same raster about the line x+y=n-1.

Exercise 5

Show that rot(z,theta1) rot(z,theta2) = rot(z,theta1 + theta2)

Exercise 6

Find the inverse of the following transformation matrix, assuming that the vectors A, B, and C are an orthonormal set. (Hint: there is a 'clean' solution).

[ Ax Bx Cx Tx ]
[ Ay By Cy Ty ]
[ Az Bz Cz Tz ]
[ 0  0  0  1  ]

CSC418 - Notes Topic 2: Basic Geometric Modeling

Key Concepts & Readings

OpenGL

Transformation Matrices

Using Transformations

Scene Graph

Notes

Coordinate-Free Geometry

2D Geometric Transforms

Affine Transformation

Derivations

Homogeneous Coordinates

Transformation Matrices

Composition and Inversion of Transformations

Comments on Matrix Composition

Hiearchical Models

Scene Graphs OR Transformation Hierarchies

Exercises

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

CSC418 - Notes
Topic 2: Basic Geometric Modeling