hw4A_98F - Solution of Homework #4, 1998 Fall MEAM 501...

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Solution of Homework #4, 1998 Fall MEAM 501 Analytical Methods in Mechanics and Mechanical Engineering 1. Bezier and B-splines cannot reproduce conics and circles. To overcome this, we shall introduce rational curves using the homogeneous coordinates ( 29 h hz hy hx P h , , , : for a point ( 29 z y x P , , : in a space. We first apply the Bezier splines to the homogeneous coordinates : ( 29 ( 29 = = + = i i i i i i i i h n i n i i h h h z h y h x h s B s r r r , 1 1 Then the corresponding curve in the three dimensional space is obtained by deviding the first three coordinates of the homogeneous ones ( 29 s h r by its homogeneous coordinate h : ( 29 ( 29 ( 29 + = + = = 1 1 1 1 n i n i i n i n i i i s B h s B h s r r . If we use the following MATHEMATICA script program : b[n_,i_,s_]:=(n!/((i-1)! (n-i+1)!)) s^(i-1) (1-s)^(n-i+1) n=3 cp={{2,1,2,2},{2,0,1,2},{2,1,2,1},{0,1,3,1}} fh=Expand[Sum[cp[[i]] b[2,i,s],{i,1,n+1}]] This is a mistake ! This line must be fh=Expand[Sum[cp[[i]] b[n,i,s],{i,1,n+1}]] f={fh[[1]],fh[[2]],fh[[3]]}/fh[[4]]
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ParametricPlot3D[f,{s,0,1}, AxesLabel->{"x","y","z"}] We have the curve Thus, the three-dimensional curve becomes as follows:
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In this example, we have the control points : i r h i r i 1 (2,1,2,2) (1,1/2,1) 2 (2,0,1,2) (1,0,1/2) 3 (2,1,2,1) (2,1,1) 4 (0,1,3,1) (0,1,3) Defining the basis function of the Bezier spline functions, and also loading a special graphic routines from MATHEMATICA:
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hw4A_98F - Solution of Homework #4, 1998 Fall MEAM 501...

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