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
– (nx,ny,d) .(x,y,1)=0 where n=(nx,ny) is unit normal
to line and d is distance from line to origin
Geometric Primitives
• 3D lines
– r = (1-l)p + lq is parametric line equation
– In this case p, q and r are 3D points
– When l = [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
– (nx,ny,nz,d) .(x,y,z,1)=0 where n=(nx,ny,nz) 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:
x' = x + t
(x ', y') = (x, y) + (t x , t y )
é 1 0 t ù
x
ê
ú
x ' = ê 0 1 ty úx
ê
ú
êë 0 0 1 úû
2D Geometric Transformations
• 2D Scaling:
x' = s × x
(x ', y') = (sx, sy)
é s 0 0
ê
x' = ê 0 s 0
ê 0 0 1
ë
ù
ú
úx
ú
û
2D Geometric Transformations
• 2D Rotation:
x' = R × x
(x ', y') = (x cosq - ysin q , x sin q + y cosq )
é cosq -sin q 0 ù
ê
ú
x' = ê sin q cosq 0 ú x
ê 0
0
1 úû
ë
2D Geometric Transformations
• 2D Rotation around arbitrary point:
– Translate to origin, rotate, translate back
x' = T(cx,cy) × R × T(-cx,-cy) × x
x ' = (x - cx)cosq - (y - cy)sin q + cx
y' = (x - cx)sinq + (y - cy)cosq + cy
é 1 0 cx ùé cosq -sin q 0 ùé 1 0 -cx ù
ê
úê
úê
ú
x' = ê 0 1 cy úê sinq cosq 0 úê 0 1 -cy ú x
ê 0 0 1 úê 0
úê 0 0 1 ú
0
1
ë
ûë
ûë
û
2D Geometric Transformations
• 2D Scaling around arbitrary point:
– Translate to origin, scale, translate back
x' = T(cx,cy) × S× T(-cx,-cy) × x
x ' = (x - cx)s + cx
y' = (x - cx)s + cy
é 1 0 cx ùé s 0 0 ùé 1 0 -cx ù
ê
úê
úê
ú
x' = ê 0 1 cy úê 0 s 0 úê 0 1 -cy ú x
ê 0 0 1 úê 0 0 1 úê 0 0 1 ú
ë
ûë
ûë
û
2D Geometric Transformations
• 2D Euclidean (rotation + translation):
– Preserves distances between points
x' = T(tx,ty) × R × x
(x ', y') = (x cosq - ysin q + t x , x sin q + y cosq + t y )
é cosq
ê
x' = ê sin q
ê
êë 0
-sin q
cosq
0
tx ù
ú
ty úx
ú
1 úû
2D Geometric Transformations
• 2D Similarity (scaling + rotation + translation):
– Preserves angles between lines
x' = T(tx,ty) × R × S× x
(x ', y') = (s x cosq - s ysinq + t x , s x sinq + s y cosq + t y )
é s × cosq
ê
x' = ê s × sinq
ê
0
êë
-s × sinq
s × cosq
0
tx ù
ú
ty úx
ú
1 úû
2D Geometric Transformations
• 2D Affine:
– Parallel lines remain parallel
x' = Ax
(x ', y') = (a00 x + a01 y + a02 , a10 x + a11 y + a12 )
é a
ê 00
x' = ê a10
ê
êë 0
a01
a11
0
a02 ù
ú
a12 ú x
ú
1 úû
2D Geometric Transformations
• 2D Projective:
– Straight lines remain straight
x' = Hx
æh x+h y+h h x+h y+h ö
01
02
11
12
(x ', y') = ç 00
, 10
÷
è h20 x + h21 y + h22 h20 x + h21 y + h22 ø
é h
ù
h
h
01
02
ê 00
ú
x' = ê h10 h11 h12 ú x
ê
ú
êë h20 h21 h22 úû
2D Geometric Transformations
• 2D Transformation hierarchy
3D Geometric Transformations
• 3D translation is very similar to 2D translation
– There are now 3 coordinates to translate (tx,ty,tz)
– 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:
x' = R × x
(x ', y', z') = (x cosq - ysinq , x sin q + y cosq , z)
é cosq -sin q 0 0 ù
ê
ú
sin q cosq 0 0 ú
ê
x' =
x
ê 0
0
1 0 ú
ê 0
ú
0
0
1
ë
û
3D Geometric Transformations
• 3D Rotation about Y axis:
x' = R × x
(x ', y', z') = (x cosq + zsin q , y, -x sin q + z cosq )
é cosq 0 sin q 0 ù
ê
ú
0
1
0
0 ú
x' = ê
x
ê -sin q 0 cosq 0 ú
ê 0
ú
0
0
1 û
ë
3D Geometric Transformations
• 3D Rotation about X axis:
x' = R × x
(x ', y', z') = (x, y cosq - zsin q , ysin q + z cosq )
é 1
0
0
0 ù
ê
ú
0 cosq -sin q 0 ú
ê
x' =
x
ê 0 sin q cosq 0 ú
ê 0
ú
0
0
1
ë
û
3D Geometric Transformations
• 3D Rotation about arbitrary axis in X-Y plane
–
–
–
–
Find angle a between rotation axis and X axis
Rotate around Z axis by -a degrees
Rotate around X axis by desired angle q
Rotate around Z axis by a 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 b between projected axis and Y axis
Rotate around X axis by -b degrees
Now rotate about axis in X-Y plane
Rotate around X axis by b 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