James obrien page 11 of 16 17 if a surface is defined

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Unformatted text preview: ________________________________ ￿ cos(# ) cos(________________________________ # ) ￿￿ dx s(x , x ) ￿ Z ||x￿ − x￿￿||2 ￿ ￿￿ cos(# ) cos(# ) x￿, x￿￿) , x￿) + "x￿(x, x￿￿)Ls(________________________________￿￿ dx ￿ − x￿￿||2 ||x S ￿ ￿￿ ￿ ￿￿ Final Exam CS 184: Foundations of Computer Graphics Fall 2009 Prof. James O’Brien page 11 of 16 17. If a surface is defined explicitly by the function x = f (u, v) , write out the equation you would use to compute the surface’s normal at some point. 4 points 18. Consider this diagram showing a four-joint arm in 2D where each joint is a simple pin joint and the base is fixed in space. If we are solving an IK problem to place the tip of the arm (the black dot) at a particular location, what is the size of the Jacobian matrix we will be working with? 3 points Draw any one configuration of the arm where two columns of the Jacobian will be parallel vectors. 5 points In the drawing you made above, clearly show the direction of the parallel vectors. 3 points In the drawing you ma...
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