1. CPT450 – Computer Graphics
Lecture 11 – Homogenous Coordinates

2. Basic Transformations
Translation by (Tx, Ty)
X’ = X + Tx
Y’ = Y + Ty
Scaling by Sx in X and Sy in Y about origin
Note: Reflection is a special case where Sx = +/-1, Sy = +/-1
X’ = X * Sx
Y’ = Y * Sy
Rotation (about origin by angle A)
X’ = X*cos(A) – Y*sin(A)
Y’ = X*sin(A) + Y*cos(A)
It would be nice if we could do all of these operations using the same equations…

3. Geometric versus Coordinate Transformations
Schaum’s Outlines on Computer Graphics (p.74) differentiates between the Geometric Transformations and the Coordinate Transformations. They are inverse functions.
Geometric transformations assume that the object is transformed and the coordinate system is fixed.
Coordinate transformations assume that the coordinate system changes and the object is fixed.
For example
Translation: Geometric transformations use (Tx, Ty) where as coordinate transformations use (-Tx, -Ty)
Scaling: Geometric transformations use Sx and Sy where as coordinate transformations use 1/Sx and 1/Sy.
Rotation: geometric use A where as coordinate transformations use (-A). Remember cos(-a) = cos(a) and sin(-a) = -sin(a)
Reflections are unchanged, due to symmetry.
Note: Sign conventions are important. If you’re not sure, use a trivial example, i.e. let the angle =90 or the scale = 2.

4. Homogeneous Coordinates
Idea: We introduce an extra coordinate, which we will set to 1.
Therefore, the 2D point [x, y] T is represented as [x,y,1] T in homogeneous coordinates.
This is equivalent to [c*x, c*y, c] T, where c is a non-zero constant. Usually it is written in normalized form, so that the last coordinate is 1.
Homogeneous coordinates allow us to express each of these transformations as a matrix multiplication.
Computers can perform these matrix calculations very quickly in software and even faster in hardware.

5. Homogenous Coordinates
Beware!
Some books (such as Schaum’s Outlines, Foley) represent points as column vectors [x,y,1] T , where T = transposed and other books (and computer graphics literature) represent points as row vectors [x,y,1].
Matrix multiplication. Note: order is different and the matrix gets transposed.
Column vector representation.
[x2,y2,1] T= [x1,y1,1] T * M T
Row vector representation
[x2,y2,1] = M * [x1,y1,1]
I will represent points as column vectors.

6. Matrix Multiplication, Vectors & Matrices
Some books do it one way, others do it the other way.
New Column Vector = Matrix * Column Vector
V’ = M x V, V = [x, y, z] T
V’(i) = M(i,1)*V(1) + M(i,2)*V(2) + M(i,3)*V(3)
New Row Vector * Matrix = new Row Vector
V’ = V x M’, V = [x, y, z]
V’(i) = V(1)*M’(1,i) + V(2)*M’(2,i) + V(3)*M’(3,i)
Note: The matrices are transposed, i.e. M’ = M T

7. Multiplying Matrices, in general
C[m,n]=A[m,l] * B[l,n], where the size of the matrices are in brackets [#rows, #cols].
Note: #cols in A = #rows in B
For i = 0 to m - 1
For j = 0 to n - 1
C(i, j) = 0
For k = 0 to l - 1
C(i, j) += A(i, k) * B(k, j)
Next k
Next j
Next i

8. Translation Matrix 1 Tx Ty

9. Matrix Multiplication
Associative
A*(B*C) = (A*B)*C
Not Commutative
A*B <> B*A
Order matters
M2 = M*X (Append)
M2 = X*M (Prepend)

10. Rotation Matrix
Cos(A)
-Sin(A)
Sin(A) 1

11. Scaling Matrix Sx Sy 1

12. Matrix Multiplication Example
Cos(90)
-Sin(90)
Sin(90) 1 10 20 1
-20 10 1 = X