04 – Geometric Transformations Overview Geometric Primitives –Points, Lines, Planes 2D Geometric Transformations –Translation, Rotation, Scaling, Affine, Projective 3D Geometric Transformations –Rotation About Arbitrary Axes

Overview Before we can understand how digital images are formed we must review some geometry –Geometric primitives –2D and 3D geometric transformations Then we will discuss how cameras create digital images of the world –3D to 2D projection –Image sampling

Geometric Primitives 2D points –x=(x,y) in Euclidean coordinates –x=(x,y,w) in homogeneous coordinates where x=x/w and y=y/w 3D points –x=(x,y,z) in Euclidean coordinates –x=(x,y,z,w) in homogeneous coordinates where x=x/w, y=y/w and z=z/w

Geometric Primitives 2D lines –ax+by+c=0 is basic 2D line equation –(a,b,c). (x,y,1)=0 using dot product notation –(n x,n y,d). (x,y,1)=0 where n=(n x,n y ) is unit normal to line and d is distance from line to origin

Geometric Primitives 3D lines –r = (1- )p + q is parametric line equation –In this case p, q and r are 3D points –When  = [0..1] we have 3D line segment

Geometric Primitives 3D planes –ax+by+cz+d=0 is basic 3D plane equation –(a,b,c,d). (x,y,z,1)=0 using dot product notation –(n x,n y,n z,d). (x,y,z,1)=0 where n=(n x,n y,n z ) is unit normal and d is distance from plane to origin

2D Geometric Transformations The simplest geometric transformations that occur in the 2D plane are: translation, Euclidean, similarity, affine and projective

2D Geometric Transformations 2D Translation:

2D Geometric Transformations 2D Scaling:

2D Geometric Transformations 2D Rotation:

2D Geometric Transformations 2D Rotation around arbitrary point: –Translate to origin, rotate, translate back

2D Geometric Transformations 2D Scaling around arbitrary point: –Translate to origin, scale, translate back

2D Geometric Transformations 2D Euclidean (rotation + translation): –Preserves distances between points

2D Geometric Transformations 2D Similarity (scaling + rotation + translation): –Preserves angles between lines

2D Geometric Transformations 2D Affine: –Parallel lines remain parallel

2D Geometric Transformations 2D Projective: –Straight lines remain straight

2D Geometric Transformations 2D Transformation hierarchy

3D Geometric Transformations 3D translation is very similar to 2D translation –There are now 3 coordinates to translate (t x,t y,t z ) –We use 4x4 matrix to perform operation 3D scaling is very similar to 2D scaling –We still only have one scale factor S –We use 4x4 matrix to perform operation

3D Geometric Transformations There are three ways to perform 3D rotations –1) Rotate about the X,Y,Z axes –2) Rotate about an arbitrary axis –3) Use unit quaternions to perform rotation Option 1 is easiest to understand Options 2 and 3 provide smoother motions

3D Geometric Transformations 3D Rotation about Z axis:

3D Geometric Transformations 3D Rotation about Y axis:

3D Geometric Transformations 3D Rotation about X axis:

3D Geometric Transformations 3D Rotation about arbitrary axis in X-Y plane –Find angle  between rotation axis and X axis –Rotate around Z axis by -  degrees –Rotate around X axis by desired angle  –Rotate around Z axis by  degrees Same approach works for rotations about an arbitrary axis in Y-Z plane or Z-X plane

3D Geometric Transformations 3D Rotation about an arbitrary axis –Project arbitrary axis onto Y-Z plane –Find angle  between projected axis and Y axis –Rotate around X axis by -  degrees –Now rotate about axis in X-Y plane –Rotate around X axis by  degrees Same approach can be used to project the arbitrary axis onto Y-Z plane or Z-X plane

3D Geometric Transformations 3D Transformation hierarchy