Geometry (Many slides adapted from Octavia Camps and Amitabh Varshney)

1
Geometry(Many slides adapted from Octavia Camps
and Amitabh Varshney)
2
Goals

• Represent points, lines and triangles as column
  vectors.
• Represent motion as matrices.
• Move geometric objects with matrix
  multiplication.
• Refresh memory about geometry and linear algebra

3
Vectors

• Ordered set of numbers (1,2,3,4)
• Example (x,y,z) coordinates of pt in space.

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Vector Addition
Vw
v
w
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Scalar Product
av
v
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Inner (dot) Product
The inner product is a SCALAR!
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How do we prove these properties of the inner
product? Lets start with the fact that
orthogonal vectors have 0 inner product. Suppose
one vector is (x,y), and WLOG x,ygt0. Then, if we
rotate that by 90 degrees counterclockwise, well
get (y, -x). Rotating the vector is just like
rotating the coordinate system in the opposite
direction. And (x,y)(y,-x) xy yx 0. Next,
note that vw (vw)/(vw) vw
This means that if we can show that when v and w
are unit vectors vw cos alpha, then it will
follow that in general vw v w cos
alpha. So suppose v and w are unit
vectors. Next, note that if w1 w2 w, then vw
v(w1w2) vw1 vw2. For any w, we can
write it as the sum of w1w2, where w1 is
perpendicular to v, and w2 is in the same
direction as v. So vw1 0. vw2 w2,
since vw2/w2 1. Then, if we just draw a
picture, we can see that cos alpha w2
vw2 vw.
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Points
Using these facts, we can represent points.
Note (x,y,z) x(1,0,0) y(0,1,0) z(0,0,1) x
(x,y,z).(1,0,0) y (x,y,z).(0,1,0) z
(x,y,z).(0,0,1)
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Lines

• Line y mx a
• Line sum of a point and a vector
• P P1 ad
• (whered is a column vector)
• Line Affine sum of two points
• P a1P1 a2P2, where a1 a2 1
• Line Segment For 0 a1, a2 1, P lies
  between P1 and P2
• Line set of points equidistant from the origin
  in the direction of a unit vector.

d
P1
P2
P1
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Plane and Triangle

• Plane sum of a point and two vectors
• P P1 au b v
• Plane set of points equidistant
• from origin in direction
• of a vector.
• Triangle Affine sum of three points
• with ai ³ 0
• P a1P1 a2P2 a3P3,
• where a1 a2 a3 1
• P lies between P1, P2, P3

v
u
P1
P3
P2
P1
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Generalizing ...

• Affine Sum of arbitrary number of points
  Convex Hull
• P a1P1 a2P2 anPn, where a1 a2
  an 1 and ai ³ 0

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Normal of a Plane

• Plane sum of a point and two vectors
• P P1 au bv
• P - P1 au bv
• Ifn is orthogonal tou andv (n u v )
• n T (P - P1) an T u bn T v 0

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Implicit Equation of a Plane

• n T (P - P1) 0
• a x
  x1
• Letn b P y P1 y1
• c z
  z1
• Then, the equation of a plane becomes
• a (x - x1) b (y - y1) c (z - z1) 0
• a x b y c z d 0
• Thus, the coefficients of x, y, z in a plane
  equation define the normal.

n
v
u
P1
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Normal of a Triangle

• Normal of the plane containing the triangle (P1,
  P2 , P3 )
• n (P2 - P1) (P3 - P1)

P3
P2
Normal pointing towards you
Normal pointing away from you
P1
P2
P1
P3

• Models constructed with consistent ordering of
  triangle vertices
• - all clockwise or all counter-clockwise.
• Usually normals point out of the model.

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Normal of a Vertex in a Mesh
nv (n1 n2 nk) / k åni / k
average of adjacent triangle normals or
better nv å (Ai ni) / (k å (Ai) )
area-weighted average of adjacent triangle
normals
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Geometry Continued

• Much of last classes material in Appendix A

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Matrices
Sum
A and B must have the same dimensions
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Matrices
Product
A and B must have compatible dimensions
Identity Matrix
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Matrices

• Associative T(U(Vp)) (TUV)p
• Distributive T(uv) Tv Tv

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Matrices
Transpose
If
A is symmetric
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Matrices
Determinant
A must be square
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Matrices
Inverse
A must be square
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Euclidean transformations
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2D Translation
P
t
P
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2D Translation Equation
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2D Translation using Matrices
P
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Scaling
P
P
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Scaling Equation
P
s.y
P
y
x
s.x
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Rotation
P
P
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Rotation Equations
Counter-clockwise rotation by an angle ?
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• Why does multiplying points by R rotate them?
• Think of the rows of R as a new coordinate
  system. Taking inner products of each points
  with these expresses that point in that
  coordinate system.
• This means rows of R must be orthonormal vectors
  (orthogonal unit vectors).
• Think of what happens to the points (1,0) and
  (0,1). They go to (cos theta, -sin theta), and
  (sin theta, cos theta). They remain orthonormal,
  and rotate clockwise by theta.
• Any other point, (a,b) can be thought of as
  a(1,0) b(0,1). R(a(1,0)b(0,1) Ra(1,0)
  Ra(0,1) aR(1,0) bR(0,1). So its in the
  same position relative to the rotated coordinates
  that it was in before rotation relative to the x,
  y coordinates. That is, its rotated.

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Degrees of Freedom
R is 2x2
4 elements
BUT! There is only 1 degree of freedom ?
The 4 elements must satisfy the following
constraints
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Transformations can be composed

• Matrix multiplication is associative.
• Combine series of transformations into one
  matrix. (example, whiteboard).
• In general, the order matters. (example,
  whiteboard).
• 2D Rotations can be interchanged. Why?

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Rotation and Translation
)
(
)
(
x
y
1
cos q -sin q tx
sin q cos q ty
0 0 1
Rotation, Scaling and Translation
)
(
)
(
x
y
1
a -b tx
b a ty
0 0 1
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Rotation about an arbitrary point

• Can translate to origin, rotate, translate back.
  (example, whiteboard).
• This is also rotation with one translation.
• Intuitively, amount of rotation is same either
  way.
• But a translation is added.

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Stretching Equation
P
Sy.y
P
y
x
Sx.x
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Stretching tilting and projecting(with weak
perspective)
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Linear Transformation
SVD
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Affine Transformation
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Viewing Position

• Express world in new coordinate system.
• If origins same, this is done by taking inner
  product with new coordinates.
• Otherwise, we must translate.

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Suppose, for example, we want to have the y axis
show how we are facing. We want to be at (7,3),
facing in direction (ct,st). The x axis must be
orthogonal, (-st,ct). If we want to express
(x,y) in this coordinate frame, we need to take
(ct,st)(x-7,y-3), and (-st,ct)(x-7,y-3). This
is done by multiplying by matrix with rows
(-st,ct) and (ct,st)
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Simple 3D Rotation
Rotation about z axis. Rotates x,y coordinates.
Leaves z coordinates fixed.
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Full 3D Rotation

• Any rotation can be expressed as combination of
  three rotations about three axes.

• Rows (and columns) of R are orthonormal vectors.
• R has determinant 1 (not -1).

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• Intuitively, it makes sense that 3D rotations
  can be expressed as 3 separate rotations about
  fixed axes. Rotations have 3 degrees of freedom
  two describe an axis of rotation, and one the
  amount.
• Rotations preserve the length of a vector, and
  the angle between two vectors. Therefore,
  (1,0,0), (0,1,0), (0,0,1) must be orthonormal
  after rotation. After rotation, they are the
  three columns of R. So these columns must be
  orthonormal vectors for R to be a rotation.
  Similarly, if they are orthonormal vectors (with
  determinant 1) R will have the effect of rotating
  (1,0,0), (0,1,0), (0,0,1). Same reasoning as 2D
  tells us all other points rotate too.
• Note if R has determinant -1, then R is a
  rotation plus a reflection.

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3D Rotation Translation

• Just like 2D case

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Transformation of lines/normals

• 2D. Line is set of points (x,y) for which
  (x,y).(ab)T0. Suppose we rotate points by R.
  We want a matrix, T, so that
• R(x,y) .T

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3D Viewing Position

• Rows of rotation matrix correspond to new
  coordinate axis.

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Rotation about a known axis

• Suppose we want to rotate about u.
• Find R so that u will be the new z axis.
• u is third row of R.
• Second row is anything orthogonal to u.
• Third row is cross-product of first two.
• Make sure matrix has determinant 1.