CSCE 441 Lecture 13

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3D Transformations as Matrices

Rotation

${\begin{bmatrix}I&o\\0&1\end{bmatrix}}\,{\begin{bmatrix}(1-\cos {\theta })\,{\vec {v}}\,{\vec {v}}^{T}+\sin {\theta }\,({\vec {v}}\times \_)+c\,I&0\\0&1\end{bmatrix}}\,{\begin{bmatrix}I&-o\\0&1\end{bmatrix}}\,{\begin{bmatrix}p\\1\end{bmatrix}}={\begin{bmatrix}{\hat {p}}\,1\end{bmatrix}}$

Mirror Image

${\begin{bmatrix}I-2{\vec {v}}\,{\vec {v}}^{T}&2{\vec {v}}\,{\vec {v}}^{T}\,o\\0&1\end{bmatrix}}\,{\begin{bmatrix}p\\1\end{bmatrix}}={\begin{bmatrix}{\hat {p}}\\1\end{bmatrix}}$

Orthogonal Projection

${\begin{bmatrix}I-{\vec {v}}\,{\vec {v}}^{T}&{\vec {v}}\,{\vec {v}}^{T}\,o\\0&1\end{bmatrix}}\,{\begin{bmatrix}p\\1\end{bmatrix}}={\begin{bmatrix}{\hat {p}}\\1\end{bmatrix}}$

Perspective Projection

Perspective transformations are not Affine!

Rational expressions require modification:

${\begin{bmatrix}\left({\vec {v}}^{T}\,(e-o)\right)\,I-e\,{\vec {v}}^{T}&e\left({\vec {v}}^{T}\,o\right)\\-{\vec {v}}^{T}&{\vec {v}}^{T}\,e\end{bmatrix}}$

homogeneous coordinates

In affine transformations, our matrix equation was

${\begin{bmatrix}L&t\\0&1\end{bmatrix}}\,{\begin{bmatrix}p\\1\end{bmatrix}}={\begin{bmatrix}{\hat {p}}\\1\end{bmatrix}}$

Now we use

${\begin{bmatrix}L&t\\c&d\end{bmatrix}}\,{\begin{bmatrix}p\\1\end{bmatrix}}={\begin{bmatrix}{\hat {p}}\\{\hat {w}}\end{bmatrix}}$

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): $\left(T\,R\right)^{-1}\,M=R^{-1}\,T^{-1}\,M$
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)