2D Today and 3D Geometric Objects and Transformations
Geometry basics Affine transformations Use of homogeneous coordinates Concatenation of transformations 3D transformations
Richard (Hao) Zhang
Introduction to Computer Graphics CMPT 361 Lecture 7
Transformation of coordinate systems Transform the transforms Transformations in OpenGL
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Why do we need transformations?
Need to transform objects from object coordinate system (OCS) to world CS Need to transform objects from world CS to the coordinate system of the camera (VCS) Need to transform objects from VCS VCS into an OpenGL window/viewport (DCS) window/viewport Object may move (translate or rotate) or deform (scale, shear, or general non-rigid deformation) nonFebruary 2, 2009
Where are we at?
Geometric transformation.
OCS WCS
DCS
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Geometry basics
Scalar, point, and vector Vector space and affine space Basic point and vector operations Sided-ness test SidedLines, planes, and triangles Linear independence Coordinate systems and frames
Scalar, point, and vector
Point: a location in space
Specified by an k-tuple for k-d points Always given with respect to some coordinate system
x P= y z
Scalar: a quantity, e.g., edge length Vector: a directed line segment between points Spaces: vector space, affine space, Euclidean space, etc.
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Vector space
A set of vectors with scalar multiplications and vector additions
Scalar-vector multiplication u = v ScalarVector-vector addition: w = u + v Vector-
Affine space
A vector space + points = affine space Operations
Vector-vector addition VectorScalar-vector multiplication ScalarPoint-vector addition PointAffine sum of points and convex sums A vector space + distance/norm = Euclidean space
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Expressions such as v = u + 2w 3r make sense in a vector space But vectors lack position
Inadequate for representing geometry we need positions, which are given by points
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Basic point and vector operations
point point = vector point + vector = point vector operations:
scalar * vector = vector vector + vector = vector vector vector = scalar, the dot product vector vector = vector, the cross product
u v v
More on dot product
u v = ux * vx + uy * vy + uz * vz u u = |u|2 is always non-negative nonu v is commutative and distributive over additions u v = |u| * |v| * cos Two vectors u and v are orthogonal if and only if u v = 0 If v is normalized, i.e., |v| = 1, then u v gives the |v 1, projection of u in the direction of v
u Right-hand rule
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More on cross product
Cross product u v is a vector perpendicular to u and v frequently used to compute the normal to a plane Direction of the cross product is determined by the right hand rule uv=vu | u v | = |u| * |v| * | sin | = area of the parallelogram How to compute? use determinant
i j k ux uy uz vx vy vz
u
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Affine and convex sums
Addition of two arbitrary points is not defined in an affine space space But consider two points P and Q with Q = P + av, we can always find av, a point R such that v = R P so now we have Q = P + a(R P) or Q = aR + (1 a)P (1 Thus, affine sum (combination) of points can be defined t1P1 + t2P2 + + tnPn, t1 + t2 + + tn = 1 Convex sum (combination) of points
u v v
t1P1 + + tnPn, where t1 + + tn = 1 and ti 0 for all i
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Convex hull
Convex hull of a set of points: set of convex combination of these points Alternatively, the convex hull is the smallest convex object containing the set of points Formed by shrink wrapping points wrapping Useful in, e.g., fast collision detection
Sided-ness test
V u
On which side does a point V lie with respect to a line, specified by a point P and a vector u?
P
Solution 1: use implicit line or plane equation; plug in the point coordinates and check the sign
u v
Solution 2: in 2D, let z coordinate be zero, compute a cross product and check the sign of the z component Solution 3: find a vector u perpendicular to u and check sign of u v u = (u , u )
y x
v P u u = (ux, uy)
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P
V
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Line representations
Consider all points of the form
P(a) = P0 + ad
Ray and line segment
If a 0, then P(a) is the ray leaving P0 in the direction of the vector d For the two-point representation twox(a) = a Rx + (1 a) Qx y(a) = a Ry + (1 a) Qy if 1 a 0, then we get all the points on the line segment joining R and Q
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this is the set of all points lying on a line that passes through P0 in the direction of the vector d Known as the parametric form of the line
Given two points R and Q on the line, we have x(a) = a Rx + (1 a) Qx y(a) = a Ry + (1 a) Qy
Other representations
Explicit: y = mx + h Implicit: ax + by + c = 0
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Parametric plane representation
A plane can be defined by a point and two vectors or by three points
P
Implicit plane equation
F(x, y, z) = ax + by + cz + d = 0 Typically, (a, b, c) is the unit normal of the plane (a If F(x0, y0, z0) < 0, point (x0, y0, z0) (x is below the plane (w. r. t. the normal) If F(x0, y0, z0) > 0, point (x0, y0, z0) (x is above the plane
(a, b, c)
v Q R u R P(a, b) = R + a(Q R) + b(P R)
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In general, the distance from a point (x0, y0, z0) to the plane is given by
ax0 + by0 + cz0 + d a 2 + b2 + c 2
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|d|
P(a, b) = R + au + bv
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Triangle and barycentric coordinates
convex sum of P and Q convex sum of S() & R
Coordinate system (CS) and frame
n linearly independent vectors of an n-D vector space define a coordinate system (CS), e.g., Cartesian CS
The vectors are called the basis vectors Any vector in the space can be written as a linear sum of the basis vectors in a unique way
An origin O, the reference point, along with a set of basis vectors form a frame
Any point P = O + linear sum of basis vectors
T is a convex sum of P, Q, and R. The weights are called the barycentric coodinates of T
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Coefficients of the sum: coordinates of point P
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From one CS to another
Express point given in one frame/CS in another CS In 2D, need to solve system of two equations
Assume the two CS have the same origin P = (1, 2)T P = (x, y)T Basis: (1, 1)T and (0, 1)T New set of basis: (2,3)T and (-1, 2)T
Change of CS
This reduces to change of basis vectors With the 2D basis vectors u (u ) and v (v ), find a 2 2 matrix M such that [u | v] M = [u | v ], thus M = [u | v]1[u | v ] v] v] Then P(u , v ) = M1P(u, v) for all points P The matrix transforms points from one CS, with basis u and v, to another CS, with basis u and v M is also called the change of basis matrix basis
Need to find x and y such that 1*(1,1)T + 2*(0,1)T = x*(2,3)T + y*(1, 2)T Alternatively
1 0 1 2 1 x 1 1 2 = 3 2 y
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Basic transformations
The most basic ones
Translation Scaling Rotation Shear And others, e.g., perspective transform, projection, etc.
Translation in 2D
P = P + T
Basic types of transformations
Rigid-body: preserves length and angle Rigid- body: Affine: preserves parallel lines, not angles or lengths Affine: Free-form: anything goes Free- form:
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x = x + d x y = y + d y
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x x d x y = y + d y
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Scaling in 2D
P = S P
x = s x x y = s y y
Uniform: sx = sy Nonuniform: sx sy
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Rotation about origin
Positive angles: counterclockwise For negative angles
sin ( ) = sin cos( ) = cos
x s x y = 0
0 x sy y
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P = R P
x cos y = sin sin x cos y
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x = x cos y sin y = x sin + y cos
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Derivation of rotation matrix
Make use of polar coordinates: (r, ) (x, y)
x = r cos y = r sin
rotate
Shearing in 2D
P = SH y P
x 1 0 x y = b 1 y
x = r cos( + ) y = r sin ( + )
y = r sin ( + ) = r sin cos + r cos sin = y cos + x sin
x = r cos( + ) = r cos cos r sin sin = x cos y sin
P = SH x P
x = x + a y x 1 a x y = 0 1 y y = y
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Homogeneous coordinates
Only translation cannot be expressed as matrix-vector matrixmultiplication But we can add a third coordinate Homogeneous coordinates for 2D points
(x, y) turns into (x, y, W) (x if (x, y, W) and (x', y', W ) are multiples of one another, they (x (x', y', represent the same point multiple representations for (x, y) (x typically, W 0. points with W = 0 are points at infinity Example for point at infinity: stereographic projection
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2D Homogeneous Coordinates
Cartesian coordinates of the homogenous point (x, y, W): x/W, y/W (divide through by W) Our typical homogenized points: (x, y, 1) (x Connection to 3D?
(x, y, 1) represents a 3D point on the plane W = 1 A homogeneous point is a line (minus origin) in 3D, through the origin Need at least one non-zero coordinate, nonso origin (0, 0, 0) is not considered
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Transformation in homogeneous coordinates
General form of affine (linear) transformation
Basic 2D transformations
1 0 d x Trans( d x , d y ) = 0 1 d y 0 0 1 cos Rot ( ) = sin 0 sin cos 0 0 0 1
sx Scale( s x , s y ) = 0 0 0 sy 0 0 0 1 0 0 1
x ' a b y ' = c d 1 0 0
p x q y 1 1
x = ax + by + p y = cx + dy + q
E.g., 2D translation in homogeneous coordinates
x ' 1 0 d x x y ' = 0 1 d y y 1 0 0 1 1
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1 a x Shear (a x ) = 0 1 0 0
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Inverse of transformations
Inverse of Trans(dx, dy) = Trans(dx, dy) Trans( Trans( d d
Compound transformations
Often we need many transforms to build objects (or to direct them) direct
e.g., rotation about an arbitrary point/line?
1 0 d x 1 0 d x 1 0 0 0 1 d 0 1 d = 0 1 0 y y 0 0 1 0 0 0 0 1 1
1 Trans(dx, dy) = Trans(dx, dy)1 Trans( d d Trans(
Concatenate basic transforms sequentially This corresponds to multiplication of the transform matrices, thanks to homogeneous coordinates
Rot()1 = Rot() Rot( 1 Rot(
1 Scale(sx, sy)1 = Scale(1/sx, 1/sy) Scale( Scale(1/s 1/s 1 Shear(ax)1 = Shear(ax) Shear( Shear(
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Compound translation
What happens when a point goes through Trans(dx1, dy1) and then Trans(dx2, dy2)? Trans( Trans( Combined translation: Trans(dx1+dx2, dy1+dy2) Trans(
Compound rotation
What happens when a point goes through Rot() and then Rot()? Rot( Rot( Combined rotation: Rot( + ) Rot(
cos sin 0 sin cos 0 0 cos 0 sin 1 0 sin cos 0 0 cos( + ) sin( + ) 0 0 = sin( + ) cos( + ) 0 1 0 0 1
1 0 d x1 1 0 d x2 1 0 d x1 + d x2 0 1 d 0 1 d = 0 1 d + d y1 y2 y1 y2 1 0 0 1 0 0 1 0 0
Concatenation of transformations: matrix multiplication
Scaling or shearing (in one direction) is similar These concatenations are commutative
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Commutativity
Transformations that do commute:
Translate translate Rotate Scale rotate scale Uniform scale rotate Shear in x (y) shear in x (y), etc.
Rotation about an arbitrary point
Break it down into easier problems: Translate to the origin: Trans(x1, y1) Trans( x y Rotate: Rot(q) Rot( And translate back: Trans(x1, y1) Trans(
In general, the order in which transformations are composed is important After all, matrix multiplications are not commutative
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Rotation about an arbitrary point
Another example
Compound transformation (non-commutative): (nonTrans(x1, y1) Rot(q) Trans(x1, y1) Trans( Rot( Trans( x y
P = Trans(x1, y1)Rot(q)Trans(x1, y1)P = TP Trans( Rot( Trans( x y Why combine these matrices into a single T?
Compound transformation: Trans(x2, y2) Rot(90o) Scale(sx, sy) Trans(x1, y1) Rot(90 Scale( Trans( x y Trans(
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Rigid-body transformation
A transformation matrix
Affine transformation (stop)
Preserves parallelism, not lengths or angles
r11 r 21 0
r12 r22 0
dx dy 1
Rot(45o)
where the upper 2 2 submatrix is orthonormal, preserves orthonormal, angles and lengths
called rigid-body transformation rigidany sequence of translation and rotation give a rigid-body rigidtransformation
Unit cube
Scale in x
Product of a sequence of translation, scaling, rotation, and shear is affine In general, A is affine if A(u) = L(u) + w, for linear transformation L and constant vector w
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Transformations in 3D (start)
Add a z-axis to (x, y) plane (x
right-handed system: right+z pointing towards us; positive rotation counter-clockwise counterleft-handed system: +z left+z pointing away from us; positive rotation clockwise
Translation in 3D
Again, we use homogeneous coordinates
Standard math convention (used Used in some graphics systems in our presentation and OpenGL) (z-axis as depth), e.g., POV-Ray, POVand Renderman
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1 0 Trans(t x , t y , t z ) = 0 0
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0 0 tx 1 0 ty 0 1 tz 0 0 1
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Scaling in 3D
sx 0 S (sx , s y , sz ) = 0 0 0 sy 0 0 0 0 sz 0 0 0 0 1
Rotation in 3D (about origin)
Around z-axis
cos sin 0 sin cos 0 Rz ( ) = 0 0 1 0 0 0 0 0 1 0 cos sin Rx ( ) = 0 sin cos 0 0 0 cos 0 sin 0 1 0 R y ( ) = sin 0 cos 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1
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Around x-axis
Around y-axis
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Properties of rotation matrix
Property 1: columns and rows are mutually orthogonal unit vectors, i.e, orthonormal r11 r12 r13 0 Property 2: determinant of M = 1 r r r 0 M = 21 22 23 r31 r32 r33 0 What if det(M) 1? 0 0 0 1 product of any pair of orthonormal matrices is also orthonormal orthormality: inverse = transpose orthormality: (PT= (P P1)
Another nice property
row vectors: unit vectors which rotate into principal axes, i.e., [1 0 0]T, [0 1 0]T, and [0 0 1]T column vectors: unit vectors into which principle axes rotate (obviously)
1 r11 0 = r 21 0 r31 r12 r22 r32 r13 r11 r23 r12 r33 r13
0 r11 1 = r 21 0 r31
r12 r22 r32
r13 r21 r23 r22 r33 r23
0 r11 0 = r 21 1 r31
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r12 r22 r32
r13 r31 r23 r32 r33 r33
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Shearing in 3D
In (y, z) w.r.t. x value
1 0 sh 1 SH yz = y shz 0 0 0 1 shx 0 1 SH xz = 0 shz 0 0 1 0 SH xy = 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 shx 0 shy 0 1 0 0 1
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Inverse Transforms
Translation: negate tx, ty, tz Scaling: change sx to 1/sx , etc. Rotation: negate the angle Shearing: negate shy, shz, etc.
In (z, x) w.r.t. y value
In (x, y) w.r.t. z value
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General 3D transformations
Any arbitrary sequence of rotation, translation scaling, and shear can be represented as:
Compound transforms
Just like in 2D, however Rotation is about more than one axis
P3 on (y, z) plane (y
r11 r M = 21 r31 0
r12 r22 r32 0
r13 r23 r33 0
tx ty tz 1
where upper left 3 3 is the combined scaling, rotation, and shearing; [tx ty tz]T for translation [t
How should we do this?
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Compute compound transform
y P3
Compute compound transform
Rotation about x by to get P1P2 to align with the +z-axis
y Ry( 90) P3 P2
Use of right-hand CS
P1 P2 x
Translation by P1 so that P1 is at origin Rotation about y by ( 90) to get P1P2 90 onto the (y, z) plane
z T(x1, y1, z1) y P3
P3
Rotation about z by to get P1P3 onto the (y, z) plane
P1 y Rx()
z
x
P3 P2 P1
P1
P2
z
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x
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z
x
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Compute compound transform
y P3 y Rz()
Alternative composition
Recall the nice properties of the rotation matrices: nice
P1
P2 x z
P3 P1 P2 x
z
r1x R = r1 y r1z
r2 x r2 y r2 z
r3 x Rx r3 y = R y r3 z Rz
Combined transformation:
Rz ( ) Rx ( ) Rot y ( 90o ) Trans ( P ) 1
Ris are the unit row vectors which rotate into principal coordinate axes, e.g., RRxT = [1 0 0]T Let us try to construct these directly, assuming the translation Trans(P1) is already done. Trans(
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Alternative composition
Alternative composition
By definition, Rz Rx must rotate into the remaining y-axis and: R y = Rz Rx We are done:
the unit vector to move to lie on the positive z axis is: the unit vector that rotates into x is normal to the plane P1P2P3.
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Rz =
Rx =
P P2 1 P P2 1
P P3 P P2 1 1 P P3 P P2 1 1
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Rx R 0 M = 1 Trans ( P ), where R = R y 0 1 Rz
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Exercise
How to get the jet into the desired direction of flight (DOF)?
Special transformations
Points: we have been doing this so far Lines: just transform the endpoint of a line Planes: trickier if defined by 3 points, can transform points, but ... more often defined by a plane equation
Ax + By + Cz + D = 0
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Plane transform
By homogeneous coordinates we can write:
N = (A B C D)
T
Plane transform: derivation
After the transform we have:
N * = QN
and we would like to have:
T
P * = MP
with P = [x y z 1]T:
NT P = 0
Now, suppose we want to transform our space by matrix M To maintain NTP = 0 , we must also transform N. Let this transform be Q.
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N * P* = 0
now some algebra:
N * P * = (QN ) (MP )
T T
= N T QT M P = 0
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(
)
Plane transform: result
This will hold when:
Q M = kI
T
Transformation of CS
So far: transform points on one object with respect to the same coordinate system (CS) Sometimes need to change CS
hence:
Q = M 1
(
)
T
e.g. we may have many objects, each in its own CS, and we want to express all of them...