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DURING THE COURSE WORK OF MY MS, I LEARN ABOUT THE ANIMATION AND THIS LECTURE SLIDES OF THIS COURSE WORK OF "Computer Animation" HAVE THE IMPORTANT POINTS:Skeletons, Matrix Review Two, Coordinate Systems, Right Handed Coordinate System, Models, Position Vectors, Model, Numverts, Components, Translation
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So we’ve basically taken the 3x3 rotation/scale matrix and the 3x1 translation vector from our original equation and combined them into a new 4x4 matrix (for today, we will always have [0 0 0 1] in the bottom row of the matrix) We also replace our 3D position vector v with its 4D version [ v x v y v z
Using 4x4 transformation matrices allows us to combine rotations, translations, scales, shears, and reflections into a single matrix For rendering, we use the bottom row as well to perform perspective projections, but for animation, we are mainly concerned with placing objects in 3D space, not rendering them into a 2D image, so we will almost always have [0 0 0 1] on the bottom row
z y x z y x z y x
θ θ θ θ
z y x r r r T r
z
Assuming we have [0 0 0 1] in the bottom row, we see that there are 12 different numbers in the upper 3x4 portion of the 4x4 matrix There are also 12 degrees of freedom for an object undergoing a linear transformation in 3D space 3 of those are represented by the three translational axes 3 of them are for rotation in the 3 planes (xy, yz, xz) 3 of them are scales along the 3 main axes and the last 3 are shears in the 3 main planes (xy, yz, xz) The 3 numbers for translation are easily decoded ( dx, dy, dz ) The other 9 numbers, however, are encoded into the 9 numbers in the upper 3x3 portion of the matrix
We will define a new space called world space or global space This space represents a 3D world or scene and may contain several objects placed in various locations Every object in the world needs a matrix that transforms its vertices from its own object space into this world space We will call this the object’s world matrix , or often, we will just call it the object’s matrix For example, if we have 100 chairs in the room, we only need to store the object space data for the chair once, and we can use 100 different matrices to transform the chair model into 100 locations in the world
We mentioned that the translation information is easily extracted directly from the matrix, while the rotation information is encoded into the upper 3x3 portion of the matrix Is there a geometric way to understand these 9 numbers? In fact there is! The 9 constants make up 3 vectors called a , b , and c. If we think of the matrix as a transformation from object space to world space, then the a vector is essentially the object’s x-axis rotated into world space, b is its y-axis in world space, and c is its z- axis in world space. d is of course the position in world space.
c a b b c a a b c a b c = × = × = × = = = 1
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