Transformations Aaron Bloomfield CS 445: Introduction to Graphics Fall 2006 (Slide set originally by David Luebke)

Graphics coordinate systems X is red Y is green Z is blue

Graphics coordinate systems If you are on the +z axis, and +y is up, then +x is to the right Math fields have +x going to the left

Outline Scaling Rotations Composing Rotations Homogeneous Coordinates Translations Projections

Scaling Scaling a coordinate means multiplying each of its components by a scalar Uniform scaling means this scalar is the same for all components:  2

Scaling Non-uniform scaling: different scalars per component: How can we represent this in matrix form? X  2, Y  0.5

Scaling Scaling operation: Or, in matrix form: scaling matrix

Outline Scaling Rotations Composing Rotations Homogeneous Coordinates Translations Projections

 2-D Rotation (x’, y’) (x, y) x’ = x cos() - y sin() y’ = x sin() + y cos()

2-D Rotation This is easy to capture in matrix form: 3-D is more complicated Need to specify an axis of rotation Simple cases: rotation about X, Y, Z axes

Rotation example: airplane

3-D Rotation What does the 3-D rotation matrix look like for a rotation about the Z-axis? Build it coordinate-by-coordinate 2-D rotation from last slide:

3-D Rotation What does the 3-D rotation matrix look like for a rotation about the Y-axis? Build it coordinate-by-coordinate

3-D Rotation What does the 3-D rotation matrix look like for a rotation about the X-axis? Build it coordinate-by-coordinate

Rotations about the axes

3-D Rotation General rotations in 3-D require rotating about an arbitrary axis of rotation Deriving the rotation matrix for such a rotation directly is a good exercise in linear algebra Another approach: express general rotation as composition of canonical rotations Rotations about X, Y, Z

Outline Scaling Rotations Composing Rotations Homogeneous Coordinates Translations Projections

Composing Canonical Rotations Goal: rotate about arbitrary vector A by  Idea: we know how to rotate about X,Y,Z So, rotate about Y by  until A lies in the YZ plane Then rotate about X by  until A coincides with +Z Then rotate about Z by  Then reverse the rotation about X (by -) Then reverse the rotation about Y (by -) Show video…

Composing Canonical Rotations First: rotating about Y by  until A lies in YZ Draw it… How exactly do we calculate ? Project A onto XZ plane Find angle  to X:  = -(90° - ) =  - 90 ° Second: rotating about X by  until A lies on Z How do we calculate ?

3-D Rotation Matrices So an arbitrary rotation about A composites several canonical rotations together We can express each rotation as a matrix Compositing transforms == multiplying matrices Thus we can express the final rotation as the product of canonical rotation matrices Thus we can express the final rotation with a single matrix!

Compositing Matrices So we have the following matrices: p: The point to be rotated about A by  Ry : Rotate about Y by  Rx  : Rotate about X by  Rz : Rotate about Z by  Rx  -1: Undo rotation about X by  Ry-1 : Undo rotation about Y by  In what order should we multiply them?

Compositing Matrices Remember: the transformations, in order, are written from right to left In other words, the first matrix to affect the vector goes next to the vector, the second next to the first, etc. This is the rule with column vectors (OpenGL); row vectors would be the opposite So in our case: p’ = Ry-1 Rx  -1 Rz Rx  Ry p

Rotation Matrices Notice these two matrices: Rx  : Rotate about X by  Rx  -1: Undo rotation about X by  How can we calculate Rx  -1? Obvious answer: calculate Rx (-) Clever answer: exploit fact that rotation matrices are orthonormal What is an orthonormal matrix? What property are we talking about?

Rotation Matrices Rotation matrix is orthogonal Columns/rows linearly independent Columns/rows sum to 1 The inverse of an orthogonal matrix is just its transpose:

Rotation Matrix for Any Axis Given glRotated (angle, x, y, z) Let c = cos(angle) Let s = sin(angle) And normalize the vector so that ||(x,y,z|| == 1 The produced matrix to rotate something by angle degrees around the axis (x,y,z) is:

Outline Scaling Rotations Composing Rotations Homogeneous Coordinates Translations Projections

Translations For convenience we usually describe objects in relation to their own coordinate system We can translate or move points to a new position by adding offsets to their coordinates: Note that this translates all points uniformly

Translation Matrices? We can composite scale matrices just as we did rotation matrices But how to represent translation as a matrix? Answer: with homogeneous coordinates

Homogeneous Coordinates Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector [x, y, z, 0]T represents a point at infinity (use for vectors) [0, 0, 0]T is not allowed Note that typically w = 1 in object coordinates

Homogeneous Coordinates Homogeneous coordinates seem unintuitive, but they make graphics operations much easier Our transformation matrices are now 4x4:

Homogeneous Coordinates Homogeneous coordinates seem unintuitive, but they make graphics operations much easier Our transformation matrices are now 4x4:

Homogeneous Coordinates Homogeneous coordinates seem unintuitive, but they make graphics operations much easier Our transformation matrices are now 4x4:

Homogeneous Coordinates Homogeneous coordinates seem unintuitive, but they make graphics operations much easier Our transformation matrices are now 4x4: Performing a scale:

More On Homogeneous Coords What effect does the following matrix have? Conceptually, the fourth coordinate w is a bit like a scale factor

More On Homogeneous Coords Intuitively: The w coordinate of a homogeneous point is typically 1 Decreasing w makes the point “bigger”, meaning further from the origin Homogeneous points with w = 0 are thus “points at infinity”, meaning infinitely far away in some direction. (What direction?) To help illustrate this, imagine subtracting two homogeneous points

Outline Scaling Rotations Composing Rotations Homogeneous Coordinates Translations Projections

Homogeneous Coordinates How can we represent translation as a 4x4 matrix? A: Using the rightmost column: Performing a translation:

Translation Matrices Now that we can represent translation as a matrix, we can composite it with other transformations Ex: rotate 90° about X, then 10 units down Z:

Translation Matrices Now that we can represent translation as a matrix, we can composite it with other transformations Ex: rotate 90° about X, then 10 units down Z:

Translation Matrices Now that we can represent translation as a matrix, we can composite it with other transformations Ex: rotate 90° about X, then 10 units down Z:

Translation Matrices Now that we can represent translation as a matrix, we can composite it with other transformations Ex: rotate 90° about X, then 10 units down Z:

Transformation Commutativity Is matrix multiplication, in general, commutative? Does AB = BA? What about rotation, scaling, and translation matrices? Does RxRy = RyRx? Does RAS = SRA ? Does RAT = TRA ?

Outline Scaling Rotations Composing Rotations Homogeneous Coordinates Translations Projections

Perspective Projection In the real world, objects exhibit perspective foreshortening: distant objects appear smaller The basic situation:

Perspective Projection When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world The view plane How tall should this bunny be?

Perspective Projection The geometry of the situation is that of similar triangles. View from above: What is x? P (x, y, z) X Z View plane d (0,0,0) x’ = ?

Perspective Projection Desired result for a point [x, y, z, 1]T projected onto the view plane: What could a matrix look like to do this?

A Perspective Projection Matrix Answer:

A Perspective Projection Matrix Example: Or, in 3-D coordinates:

A Perspective Projection Matrix OpenGL’s gluPerspective() command generates a slightly more complicated matrix: Can you figure out what this matrix does?

Projection Matrices Now that we can express perspective foreshortening as a matrix, we can composite it onto our other matrices with the usual matrix multiplication End result: can create a single matrix encapsulating modeling, viewing, and projection transforms Though you will recall that in practice OpenGL separates the modelview from projection matrix (why?)