CSCE 441 Lecture 13

« previous | Wednesday, February 12, 2014 | next »

3D Transformations as Matrices

Rotation

$\left[\begin{array}{cc}I& o\\ 0& 1\end{array}\right]\left[\begin{array}{cc}(1-\cos \theta )\overrightarrow{v}{\overrightarrow{v}}^T+\sin \theta (\overrightarrow{v}\times \mathrm{\_})+cI& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}I& -o\\ 0& 1\end{array}\right]\left[\begin{array}{c}p\\ 1\end{array}\right]=\left[\begin{array}{c}\hat{p}1\end{array}\right]$

Mirror Image

$\left[\begin{array}{cc}I-2\overrightarrow{v}{\overrightarrow{v}}^T& 2\overrightarrow{v}{\overrightarrow{v}}^{T}o\\ 0& 1\end{array}\right]\left[\begin{array}{c}p\\ 1\end{array}\right]=\left[\begin{array}{c}\hat{p}\\ 1\end{array}\right]$

Orthogonal Projection

$\left[\begin{array}{cc}I-\overrightarrow{v}{\overrightarrow{v}}^T& \overrightarrow{v}{\overrightarrow{v}}^{T}o\\ 0& 1\end{array}\right]\left[\begin{array}{c}p\\ 1\end{array}\right]=\left[\begin{array}{c}\hat{p}\\ 1\end{array}\right]$

Perspective Projection

Perspective transformations are not Affine!

Rational expressions require modification:

$\left[\begin{array}{cc}({\overrightarrow{v}}^T(e-o))I-e{\overrightarrow{v}}^T& e\left({\overrightarrow{v}}^{T}o\right)\\ -{\overrightarrow{v}}^T& {\overrightarrow{v}}^{T}e\end{array}\right]$

homogeneous coordinates

In affine transformations, our matrix equation was

$\left[\begin{array}{cc}L& t\\ 0& 1\end{array}\right]\left[\begin{array}{c}p\\ 1\end{array}\right]=\left[\begin{array}{c}\hat{p}\\ 1\end{array}\right]$

Now we use

$\left[\begin{array}{cc}L& t\\ c& d\end{array}\right]\left[\begin{array}{c}p\\ 1\end{array}\right]=\left[\begin{array}{c}\hat{p}\\ \hat{w}\end{array}\right]$

Where $c$ and $d$ represent the stuff in the denominator of the transformation equation

3D Location is $\frac{\hat{p}}{\hat{w}}$

Note this equation still works for Affine transformations (where we divide by 1)

Hierarchical Animation

Multiple transformations at each joint of a skeleton.

For example, to move a point on the hand, we have to move the wrist ($M_1$), elbow ($M_2$), and shoulder ($M_3$).

$\hat{p}=M_3{M}_2{M}_{1}p$

In general,

$\hat{p}=\sum _j{\alpha }_j{M}_{j}p$

Transformations in OpenGL

4 types:

• View
• Model (put things in the scene)
• Projection (takes 3D objects and projects onto a 2D surface)
• Viewport

Commands:

• glTranslatef(x,y,z)
• glScalef(x,y,z)
• glRotatef(theta, vx, vy, vz)
• glLoadIdentity(void): replaces current matrix with identity matrix (i.e. removes current transformations)
• glPushMatrix(void) and glPopMatrix(void): copies current matrix to new context and removes current matrix (respectively)

Order matters!

• glTranslatef(0,1,3) = M[1]
• glRotatef(45,0,1,0) = M[2]
• glScalef(1,1,2) = M[3]

$M_1{M}_2{M}_{3}P$

OpenGL combines model and view matrix:

$ModelView=V^{-1}M$

1. push matrix
2. specify viewer using inverse of view transformation (negative translation/rotation and in opposite order): ${\left(TR\right)}^{-1}M=R^{-1}{T}^{-1}M$
3. push matrix
4. position objects with transformations in opposite order of what you would expect
5. pop matrix
6. repeat last 3 steps for each object in scene
7. pop matrix

Special Matrix Commands

• glMatrixMode(GL_MODELVIEW) and glMatrixMode(GL_PROJECTION)
• gluLookAt(ex,ey,ez, cx,cy,cz, ux,uy,uz): set position of eye, direction of thing to look at, and the "up direction"
• gluPerspective(fov, aspect, near, far)
• glOrtho(left, right, bottom, top, near far) specify orthogonal projection
• glViewport(x,y,w,h)