Week 4 - Friday What did we talk about last time? Homogeneous - - PowerPoint PPT Presentation

Week 4 - Friday

 What did we talk about last time?  Homogeneous notation  Vector equations for lines and planes

 Once we are in 3D, we have to talk about planes as well  The explicit form of a plane is similar to a line:

• p(u,v) = o + us + vt
• o is a point on the plane
• s and t are vectors that span the plane
• s x t is the normal of the plane

 The implicit equation for a plane is just like the implicit for a line,

except with an extra dimension

 Let n be a vector (a, b, c)  Let p be a point (x, y, z)  p is on plane π if and only if

• n • p + d = 0

 Since n is the normal of the plane we can compute it in a couple of

ways:

• n = s x t where s and t span the plane
• n = (u – w) x (v – w) where u, v, and w are noncollinear points in the plane

 Again, it works just like a line  If we write f(p) = n • p + d

 Assuming q ∈ plane π

• f(p) = 0 iff p ∈ π
• f(p) > 0 iff p lies on the same side as the point q + n
• f(p) < 0 iff p lies on the same side as the point q - n

 f(p) again gives a (scaled) measure of the perpendicular distance

from p to π

 Distance(p) = f(p)/||n||  This implies that Distance(0) = d, making d the shortest distance

from the origin to the plane

 The cross product of two vectors finds a vector that is

• rthogonal to both

 For 3D vectors u and v in an orthonormal basis, the cross

product w is:

          − − − = × =           =

x y y x z x x z y z z y z y x

v u v u v u v u v u v u w w w v u w

 Plane defined by:

• (1, 3, 2)
• (2, 5, 2)
• (3, 8, 2)

 Where is (3, 3, 2)?  What about (3, 4, 5)?

 Let p = (px, py) be a unit vector  Let ϕ be the angle the vector makes with the x-axis  sin ϕ = py  cos ϕ = px  tan ϕ = sin ϕ/cos ϕ = py/px

 For a right triangle with sides a and b and hypotenuse c:

• sin α = a/c
• cos α = b/c
• tan α = sin α/cos α = a/b
• c2 = a2 + b2

 For any triangle:

• Law of sines:
• Law of cosines:

c2 = a2 + b2 – 2ab cosγ

• Law of tangents:

c γ b β a α sin sin sin = =

2 tan 2 tan β α β α b a b a − + = − +

 Trigonometric identity: cos2ϕ + sin2ϕ = 1  Double angle:

• sin2 ϕ = 2sin ϕcos ϕ = 2tan ϕ/(1 + tan2 ϕ)
• cos2 ϕ=cos2ϕ- sin2ϕ=1 – sin2ϕ
• tan2 ϕ = 2tan ϕ/(1-tan2 ϕ)

 There are a host of other useful ones listed in the book

 The computer science behind texture mapping uses a lot of

techniques to make it look better

 For now, we're only worried about mapping a vertex to a

location in a rectangular texture

• MonoGame will take care of the rest

 Texture coordinates are usually

given in (u, v) form

 u is the horizontal direction and

v is the vertical

 These coordinates are each

almost always in the range [0, 1]

 In MonoGame, u grows from left

to right and v grows from top to bottom

u v

(0,0) (1,0) (0,1)

 Load images into Texture2D objects as before  Use a vertex format that allows a texture to be specified (such

as VertexPositionNormalTexture)

 Set appropriate texture coordinates on each vertex  When drawing, set the Texture property on the

BasicEffect object to the texture you want to use

 Apply the change on the pass

 A transform is an operation that changes points, vectors, or

colors

 We can use them to position and animate objects, lights, and

cameras

 A linear transform is one that holds over vector addition and

scalar multiplication

• Rotation
• Scaling
• Can be represented by a 3 x 3 matrix

 Adding a vector after a linear transform makes an affine

transform

 Affine transforms can be stored in a 4 x 4 matrix using

homogeneous notation

 Affine transforms:

• Translation
• Rotation
• Scaling
• Reflection
• Shearing

Notation Name Characteristics T(t) Translation matrix Moves a point (affine) R Rotation matrix Rotates points (orthogonal and affine) S(s) Scaling matrix Scales along x, y, an z axes according to s (affine) Hij(s) Shear matrix Shears component i by factor s with respect to component j E(h,p,r) Euler transform Orients by Euler angles head (yaw), pitch, and roll (orthogonal and affine) Po(s) Orthographic projection Parallel projects onto a plane or volume (affine) Pp(s) Perspective projection Project with perspective onto a plane or a volume slerp(q,r,t) Slerp transform Interpolates quaternions q and r with parameter t

 More on standard transforms  Normal transforms  Inverses  Euler transform  Matrix decomposition  Rotation around an arbitrary axis  Quaternions

 Keep reading Chapter 4  Work on Project 1, due next Friday