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04-3D_transformations.md

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3D Transformations

The Camera Analogy

  • Viewing Transformation — setting up tripod and pointing camera
  • Modelling Transformation — arranging the scene
  • Projection Transformation — adjusting lens or zoom
  • Viewport Transformation — how large you want the photo to be
  • Given a set of coordinates v and transformation M, then v' = Mv

Viewing & Modelling Transformation

  • modelview(GL_MODELVIEW) matrix — combined viewing and modelling transformations, applied to incoming object coordinates to yield eye coordinates
  • glLoadIdentity() matrix — called to reset the ModelView matrix
  • gluLookAt(GLdouble eyex, GLdouble eyey, GLdouble eyez, GLdouble centerx, GLdouble centery, GLdouble centerz, GLdouble upx, GLdouble upy, GLdouble upz) — viewing transformation function
    • e.g. gluLookAt(0.0, 0.0, 5.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0)
  • glTranslatef(x, y, z) — used to model/move the objects in the scene

Projection Transformation

  • Selecting using GL_PROJECTION
    • glMatrixMode(GL_PROJECTION)
    • glLoadIdentity()

Perspective Projection

  • glFrustrum(left, right, bottom, top, near, far) calculates a matrix that specifies the perspective viewing volume
  • gluPerspective(fovy, aspect, near, far) is the same as above, but uses aspect ratio to specific the viewbox and the angle (fovy) of viewing

Orthographic Projection

  • glOrtho(left, right, bottom, top, near, far) calculates a viewing matrix where distance of objects from a camera doesn't affect its perceived size

Viewport Transformation

  • Viewport defines the size and location of the final processed image in the application window
  • glViewPort(x, y, width, height) specifies the viewport
  • reshape() is called to assign new width and height variables on window resize

Example

public void reshape(GLAutoDrawable d, int x, int y, int w, int h) {
	GL2 gl = d.getGL().getGL2();
	
	if (h == 0) h = 1;
	float aspect = (float) w / h;
	
	gl.glViewport(0,0, w, h);
	
	gl.glMatrixMode(GL_PROJECTION);
	gl.glLoadIdentity();
	glu.gluPerspective(45.0, aspect, 0.1, 100.0);
	
	gl.glMatrixMode(GL_MODELVIEW);
	gl.glLoadIdentity();

}

3D Transformation

Translation

  • x' = x + ∆x
  • y' = y + ∆y
  • z' = z + ∆z
  • w' = w
  • And expressed as a vector:
|x'|   |1 0 0 ∆x| |x|
|y'| = |0 1 0 ∆y| |y|
|z'|   |0 0 1 ∆z| |z|
|w'|   |0 0 0 1 | |w|

Scaling

  • x' = ax
  • y' = by
  • z' = cz
  • w' = w
  • And expressed as a vector:
|x'|   |a 0 0 0| |x|
|y'| = |0 b 0 0| |y|
|z'|   |0 0 c 0| |z|
|w'|   |0 0 0 1| |w|

Shearing

  • tanθ = y/∆x → cotθ = ∆x/y
  • x' = x + ycotθ
  • y' = y
  • z' = z
  • w' = w
  • And in vector notation:
|x'|   |1 cotθ 0 0| |x|
|y'| = |0 1    0 0| |y|
|z'|   |0 0    1 0| |z|
|w'|   |0 0    0 1| |w|

Rotation

  • x' = rcos(θ + ɸ) = xcosθ - ysinθ
  • y' = rsin(θ + ɸ) = ycosθ + xsinθ
  • z' = z
  • w' = w
  • glRotate*(θ, 0, 0, 1) for z-axis
  • glRotate*(θ, 1, 0, 0) for x-axis
  • glRotate*(θ, 0, 1, 0) for y-axis
  • To rotate about an arbitrary point (x,y),
    • it must be transformed to origin
    • rotated
    • then transformed back to its original position
  • And expressed as a vector:
Z ROTATION:
|x'|   |cosθ -sinθ 0 0| |x|
|y'| = |sinθ cosθ  0 0| |y|
|z'|   |0    0     1 0| |z|
|w'|   |0    0     0 1| |w|

X ROTATION:
|x'|   |1 0     0     0| |x|
|y'| = |0 cosθ  -sinθ 0| |y|
|z'|   |0 sinθ  cosθ  0| |z|
|w'|   |0 0     0     1| |w|

Y ROTATION:
|x'|   |cosθ  0 sinθ 0| |x|
|y'| = |0     1 0    0| |y|
|z'|   |-sinθ 0 cosθ 0| |z|
|w'|   |0     0 0    1| |w|

Clearing Color and Depth Buffer

  • gl.glClearColor(r,g,b,a)
  • gl.glClear(GL2.GL_COLOR_BUFFER_BIT)
  • Or for both: gl.glClear(GL2.GL_COLOR_BUFFER_BIT | GL2.GL_DEPTH_BUFFER_BIT)