DIFFERENTIAL GEOMETRY OF SURFACES 199
- Develop a procedure for viewing surface geometry. Your program should display a parametric surface by
drawing iso-parametric curves and surface geometry moving along any such curve picked by the user. The
following geometric entities should be displayable: (i) two partial derivatives, (ii) unit surface normal and
(iii) tangent plane at the point.
- Given a bi-cubic patch
r( , ) = [ 1]
[1 2 0] [1 4 4] [0 2 4] [0 2 4]
[4 6 2] [4 8 6] [0 2 4] [0 2 4]
[5 2 –2] [5 2 –2] 0 0
[3 4 2] [3 4 2] 0 0 1
321
3
2
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v
v
v
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Determine whether it is a developable surface.
- A bi-cubic patch is given by
r( , ) = [ 1]
[0 0 10] [10 0 0] [16 0 0] [0 0 –16]
[0 10 8] [18 10 0] [24 0 0] [0 0 –14]
[0 10 –2] [8 10 0] [8 0 0] [0 0 2]
[0 10 –2] [8 10 0] [8 0 0] [0 0 2] 1
32
3
2
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v
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Determine the
(a) coordinates on the surface at P (0.5, 0.5)
(b) unit normal at P
(c) unit tangent at P
(d) equation of the tangent plane at P
(e) Gaussian quadrature at P
- Prove the Weingarten relations
H^2 nu = (G 12 M–G 22 L)ru + (G 12 L–G 11 M)rv
H^2 nv = (G 12 N – G 22 M)ru + (G 12 M–G 11 N)rv
and show that
H(nu×nv) = (LN–M^2 )n
1
0
–1
1
0
–1
0.5
0.25
0
Figure P6.2 Mobius strip
