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 + u s + v t  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 orthogonal to both  For 3D vectors u and v in an orthonormal basis, the cross product w is:   −   w u v u v     x y z z y = = × = −     w w u v u v u v y z x x z     −  w  u v u v   z x y y x

 Plane defined by:  (1, 3, 2)  (2, 5, 2)  (3, 8, 2)  Where is (3, 3, 2)?  What about (3, 4, 5)?

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

 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  c 2 = a 2 + b 2

 For any triangle: sin α sin β sin γ = =  Law of sines: a b c c 2 = a 2 + b 2 – 2 ab cosγ  Law of cosines: + α β tan +  Law of tangents: a b 2 = − − α β a b tan 2

 Trigonometric identity: cos 2 ϕ + sin 2 ϕ = 1  Double angle:  sin2 ϕ = 2sin ϕ cos ϕ = 2tan ϕ /(1 + tan 2 ϕ )  cos2 ϕ =cos 2 ϕ - sin 2 ϕ =1 – sin 2 ϕ  tan2 ϕ = 2tan ϕ /(1-tan 2 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 u (0,0) (1,0) given in ( u , v ) form  u is the horizontal direction and v is the vertical  These coordinates are each v almost always in the range [0, 1]  In MonoGame, u grows from left to right and v grows from top to bottom (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) H ij ( 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) P o ( s ) Orthographic Parallel projects onto a plane or volume (affine) projection P p ( 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