CS 480/CS680 PSet 3: Curves & Surfaces
1. (20 points; 5 points extra credit)
We are given the implicit function for a two-sheeted hyperboloid,
๐(๐ฅ, ๐ฆ, ๐ง) = (๐ฅ โ ๐เฏซ
)
เฌถ
+ เตซ๐ฆ โ ๐เฏฌเตฏ
เฌถ โ ๐ง
เฌถ + ๐
เฌถ
.
a) Give the 4 ร 4 matrix ๐ธ for this cylindrical surface, such that ๐
๐ป๐ธ๐ = 0 for any
homogeneous point ๐ = (๐ฅ, ๐ฆ, ๐ง, 1) that is on the cylindrical surface.
b) Using the implicit function, derive a function that gives a unit normal vector at any point
on the surface (๐ฅ, ๐ฆ, ๐ง).
c) Give an equivalent parametric equation for the sheet of the surface that is above the xyplane (๐ง โฅ 0), in terms of ฮธ and z. Assume that โ๐ โค ๐ โค ๐ and ๐งเฏ เฏเฏก โค ๐ง โค ๐งเฏ เฏเฏซ.
Give the parameters for ๐งเฏ เฏเฏก, ๐งเฏ เฏเฏซ.
d) Using the parametric equation, derive the function that gives the unit normal vector at
any point on the sheet with parameter (๐, ๐ง).
e) Extra Credit (5 points): Give a parametrization that utilizes hyperbolic trigonometric
functions, and find the unit normal with this parametrization.
2. (30 points)
We are given a 3D swept surface. A line segment with end points ๐๐ = (2,0,0) and
๐๐ = (0,0,6) is used as the sweep curve. The cross-section is an ellipse in the z = 0 plane,
where ๐ฅ = 2 + ๐เฏซ ๐๐๐ (2๐๐ฃ), ๐ฆ = ๐เฏฌ ๐ ๐๐(2๐๐ฃ), 0 โค ๐ฃ โค 1. The center of the ellipse is
swept along the vector from ๐๐ to ๐๐ , such that it remains parallel to the x-y plane.
a) Derive the parametric equation in ๐ข for the directed line segment ๐๐ ๐๐.
b) Derive an equation for location of the point ๐ท(๐ข, ๐ฃ) on the swept surface.
c) Derive the equation for the swept surface normal ๐ต(๐ข, ๐ฃ).
3. (20 points)
We are given a 2D cubic Bezier curve segment, which has the following control points:
๐๐ = (2, 3)
๐๐ = (3, 0)
Submission guidelines:
Please prepare your answers neatly written or typed, on separate 8.5โx11โ sheets. Submit on
Blackboard. If your answers are hand-written please either upload a scan or photos of your
answer sheets.
๐๐ = (4, 4)
๐๐ = (5, 1)
a) Draw the convex hull for this 2D Bezier curve segment.
b) Compute the value of ๐โ(0) for this 2D Bezier curve segment.
c) We are now given a second 2D Bezier curve segment, which has the control points:
๐๐ = (0, -1)
๐๐ = (-2, 4)
๐๐ = (0, 9)
๐๐ = (2, 3)
Does this segment join the previous segment with C1 continuity? Give a mathematical
justification for your answer.
d) Which control point above (for the second curve in (c)) may we change to achieve C1
continuity? Write down the new position of the point that will achieve this.
4. (30 points)
We are given the following boundary conditions for a cubic spline section:
๐ท(0) = ๐เฏ
๐ท(1) = ๐เฏเฌพเฌต
๐ทโฒ(0) = เฌต
เฌถ
[(1 + ๐)(๐เฏ โ ๐เฏเฌฟเฌต) + (1 โ ๐)(๐เฏเฌพเฌต โ ๐เฏ
)]
๐ทโฒ(1) =
เฌต
เฌถ
[(1 + ๐)(๐เฏเฌพเฌต โ ๐เฏ
) + (1 โ ๐)(๐เฏเฌพเฌถ โ ๐เฏเฌพเฌต)]
In the textbook, we see this is a Cardinal Spline (Kochanek-Bartels spline with t=0 and c=0).
In this case ๐เฏเฏเฏขเฏ = [๐เฏเฌฟเฌต ๐เฏ ๐เฏเฌพเฌต ๐เฏเฌพเฌถ]
เฏ
and the boundary conditions can be written:
โฃ
โข
โข
โก
๐ท(0)
๐ท(1)
๐ทโฒ(0)
๐ทโฒ(1)โฆ
โฅ
โฅ
โค
=
โฃ
โข
โข
โก
0 1 0 0
0 0 1 0
เฐท
เฐญ
เฐฎ
(เฌตเฌพเฏ) ๐
เฐญ
เฐฎ
(เฌตเฌฟเฏ) 0
0 เฐท
เฐญ
เฐฎ
(เฌตเฌพเฏ) ๐
เฐญ
เฐฎ
(เฌตเฌฟเฏ)โฆ
โฅ
โฅ
โค
เตฆ
๐เฏเฌฟเฌต
๐เฏ
๐เฏเฌพเฌต
๐เฏเฌพเฌถเตช
a) Show how to compute the 4ร4 coefficient matrix MC given the boundary conditions
written above. You do not need to compute a matrix inverse to find MC (the relevant one
is given in the textbook anyway). Just give the specific equations for MC.
b) Given MC write out the blending functions for this curve.
c) Do adjacent segments satisfy C1 continuity? Give a mathematical justification.
d) Does adjusting b change the tangent direction at the endpoints or only magnitude?