人机接口与图形学(双语)03-Transformations.ppt

Looking At Quaternions But interpolating multiple rotations is still ugly Quaternions have some other nice advantages too more compact than rotation matrices can compose rotations by quaternion multiplication but they can be easily converted to matrices if needed How Does SLERP Work ? Given two object poses from two rotation matrices M and M’; Convert M to quaternion q, M’ to quaternion q’; Obtain the new quaternion q’/q with a new rotation axis A and angle Q; How Does SLERP Work ? Uniformly subdivide the rotation angle Q into a few smaller angles and compose intermediate quaternions using axis A; Convert the intermediate quaternions back to intermediate rotation matrices; How Does SLERP Work ? Apply intermediate rotation matrices to the first object pose to generate intermediate poses towards the second pose. Transformation of Normal Vectors Affine transformations map parallel lines to parallel lines but the same does not hold for perpendicular lines Transform M will not map normal vectors to normal vectors first guess would be to map normals as n → Mn after transform, may or may not be perpendicular to surface Normal vectors are defined by surface tangent planes so let’s consider how planes are transformed Transformation of Normal Vectors A plane in 3-D space is described by the homogeneous vector thus any point v on the plane satisfies the equation Transformation of Normal Vectors For any 4x4 matrix whose inverse exists, this is equivalent to thus the transformed point Mv lies on the plane It’s plane vector is Transformation of Normal Vectors Transformation of Normal Vectors This gives us the transformation rule for normal vectors Transformation of Normal Vectors Must in general compute actual local plane however, there are some simpler cases Simplified case #1: Affine Transformations map parallel planes to parallel planes thus, can pick any value of d — might as well be 0 Transformation of Normal Vectors Simplified case #2: Orthogo