Skeletons CSE169: Computer Animation Instructor: Steve Rotenberg - PowerPoint PPT Presentation

Skeletons CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2020

Matrix Review

Coordinate Systems ◼ Right handed coordinate system y x z

3D Models ◼ Let’s say we have a 3D model that has an array of position vectors describing its shape ◼ We will group all of the position vectors used to store the data in the model into a single array: v n where 0 ≤ n ≤ NumVerts-1 ◼ Each vector v n has components v nx v ny v nz

Translation ◼ Let’s say that we want to move our 3D model from its current location to somewhere else… ◼ In technical jargon, we call this a translation ◼ We want to compute a new array of positions v ′ n representing the new location ◼ Let’s say that vector d represents the relative offset that we want to move our object by ◼ We can simply use: v ′ n = v n + d to get the new array of positions

Transformations v ′ n = v n + d ◼ This translation represents a very simple example of an object transformation ◼ The result is that the entire object gets moved or translated by d ◼ From now on, we will drop the n subscript, and just write v ′ = v + d remembering that in practice, this is actually a loop over several different v n vectors applying the same vector d every time

Transformations  = + v v d ◼ Always remember that this compact equation can be expanded out into        v v d x x x        = + v v d       y y y              v v d z z z ◼ Or into a system of linear equations:  = + v v d x x x  = + v v d y y y  = + v v d z z z

Rotation ◼ Now, let’s rotate the object in the xy plane by an angle θ , as if we were spinning it around the z axis ( ) ( )  =  −  v cos v sin v x x y ( ) ( )  =  +  v sin v cos v y x y  = v v z z ◼ Note: a positive rotation will rotate the object counterclockwise when the rotation axis (z) is pointing towards the observer

Rotation ( ) ( )  =  −  v cos v sin v x x y ( ) ( )  =  +  v sin v cos v y x y  = v v z z We can expand this to: ◼ ( ) ( )  =  −  + v cos v sin v 0 v x x y z ( ) ( )  =  +  + v sin v cos v 0 v y x y z  = + + v 0 v 0 v 1 v z x y z And rewrite it as a matrix equation: ◼   −        v cos sin 0 v x x        =    v sin cos 0 v       y y              v 0 0 1 v z z Or just: ◼  =  v M v

Rotation ◼ We can represent a z-axis rotation transformation in matrix form as:   −        v cos sin 0 v x x        =    v sin cos 0 v       y y              v 0 0 1 v z z or more compactly as:  =  v M v where  −    cos sin 0   ( ) =  =   M R sin cos 0   z     0 0 1

Rotation We can also define rotation matrices for the x , y , and z axes: ◼   1 0 0   ( )  =  −  R 0 cos sin   x       0 sin cos     cos 0 sin   ( )  = R 0 1 0   y   −     sin 0 cos  −    cos sin 0   ( )  =   R sin cos 0   z     0 0 1

Linear Transformations ◼ Like translation, rotation is an example of a linear transformation ◼ True, the rotation contains nonlinear functions like sin()’s and cos()’s, but those ultimately just end up as constants in the actual linear equation ◼ We can generalize our matrix in the previous example to be:   a b c 1 1 1    =  = M a b c v M v   2 2 2     a b c 3 3 3

Linear Equation ◼ A general linear equation of 1 variable is: ( ) = + f v av d where a and d are constants ◼ A general linear equation of 3 variables is: ( ) ( ) = = + + + f v , v , v f v av bv cv d x y z x y z 2 , ◼ Note: there are no nonlinear terms like v x v y , v x sin( v x )…

System of Linear Equations ◼ Now let’s look at 3 linear equations of 3 variables v x , v y , and v z  = + + + v a v b v c v d x 1 x 1 y 1 z 1  = + + + v a v b v c v d y 2 x 2 y 2 z 2  = + + + v a v b v c v d z 3 x 3 y 3 z 3 ◼ Note that all of the a n , b n , c n , and d n are constants (12 in total)

Matrix Notation  = + + + v a v b v c v d x 1 x 1 y 1 z 1  = + + + v a v b v c v d y 2 x 2 y 2 z 2  = + + + v a v b v c v d z 3 x 3 y 3 z 3          v a b c v d x 1 1 1 x 1          =  + v a b c v d         y 2 2 2 y 2                  v a b c v d z 3 3 3 z 3  =  + v M v d

Translation ◼ Let’s look at our translation transformation again:  = + v v d x x x  = +  = + v v d v v d y y y  = + v v d z z z ◼ If we really wanted to, we could rewrite our three translation equations as:  = + + + v 1 v 0 v 0 v d x x y z x  = + + + v 0 v 1 v 0 v d y x y z y  = + + + v 0 v 0 v 1 v d z x y z z

Identity ◼ We can see that this is equal to a transformation by the identity matrix  = + + + v 1 v 0 v 0 v d x x y z 1  = + + + v 0 v 1 v 0 v d y x y z 2  = + + + v 0 v 0 v 1 v d z x y z 3          v 1 0 0 v d x x 1          =  + v 0 1 0 v d         y y 2                  v 0 0 1 v d z z 3

Identity ◼ Multiplication by the identity matrix does not affect the vector   1 0 0   = I 0 1 0       0 0 1 =  v I v

Uniform Scaling We can apply a uniform scale to our object with the following ◼ transformation        v s 0 0 v x x        =  v 0 s 0 v       y y              v 0 0 s v z z If s >1, then the object will grow by a factor of s in each dimension ◼ If 0< s <1, the object will shrink ◼ If s<0, the object will be reflected across all three dimensions, ◼ leading to an object that is ‘inside out’

Non-Uniform Scaling ◼ We can also do a more general nonuniform scale , where each dimension has its own scale factor        v s 0 0 v x x x        =  v 0 s 0 v       y y y              v 0 0 s v z z z which leads to the equations:  = v s v x x x  = v s v y y y  = v s v z z z

Reflections ◼ A reflection is a special type of scale operation where one of the axes is negated causing the object to reflect across a plane ◼ For example, a reflection along the x-axis would look like: ◼ Given an arbitrary 3x3 matrix M , we can tell if it has been reflected by checking if the matrix determinant is negative

Shears ◼ A shear is a translation along one axis by an amount proportional to the value along a different axis ◼ It causes a deformation similar to writing with italics (i.e., it causes a rectangle to deform into a parallelogram) ◼ For example a shear along x proportional to the value of y would look like:

Multiple Transformations ◼ If we have a vector v , and an x-axis rotation matrix R x , we can generate a rotated vector v ′: ( ) v  =   v R x ◼ If we wanted to then rotate that vector around the y-axis, we could simply: ( )    =   v R v y ( ) ( ( ) )   =     v R R v y x

Multiple Transformations We can extend this to the concept of applying any sequence of ◼ transformations: ( ( ( ) ) )  =     v M M M M v 4 3 2 1 Because matrix algebra obeys the associative law, we can regroup this as: ◼ ( ) v  =     v M M M M 4 3 2 1 This allows us to concatenate them into a single matrix: ◼ =    M M M M M total 4 3 2 1  =  v M v total Note: matrices do NOT obey the commutative law, so the order of ◼ multiplications is important