CS680-PSet 3: Curves & Surfaces Solved

We are given the implicit function for a two-sheeted hyperboloid,  
                                                          𝑓(𝑥, 𝑦, 𝑧) = (𝑥 − 𝑐 ) + 𝑦 − 𝑐           − 𝑧    + 𝑟 .  

Give the 4 × 4 matrix 𝑸 for this cylindrical surface, such that 𝒑 𝑻𝑸𝒑 = 0 for any homogeneous point 𝑝 = (𝑥, 𝑦, 𝑧, 1) that is on the cylindrical surface.
Using the implicit function, derive a function that gives a unit normal vector at any point on the surface (𝑥, 𝑦, 𝑧).
Give an equivalent parametric equation for the sheet of the surface that is above the xyplane (𝑧 ≥ 0), in terms of θ and z. Assume that −𝜋 ≤ 𝜃 ≤ 𝜋 and 𝑧 ≤ 𝑧 ≤ 𝑧 . Give the parameters for 𝑧        , 𝑧         .
Using the parametric equation, derive the function that gives the unit normal vector at any point on the sheet with parameter (𝜃, 𝑧).
Extra Credit (5 points): Give a parametrization that utilizes hyperbolic trigonometric functions, and find the unit normal with this parametrization.
We are given a 3D swept surface. A line segment with end points 𝒑𝟎 = (2,0,0) and  𝒑𝟏 = (0,0,6) is used as the sweep curve. The cross-section is an ellipse in the z = 0  plane, where 𝑥 = 2 + 𝑟 𝑐𝑜𝑠(2𝜋𝑣), 𝑦 = 𝑟 𝑠𝑖𝑛(2𝜋𝑣), 0 ≤ 𝑣 ≤ 1. The center of the ellipse is swept along the vector from  𝒑𝟎  to  𝒑𝟏 , such that it remains parallel to the x-y plane.

Derive the parametric equation in 𝑢 for the directed line segment 𝒑𝟎 𝒑𝟏.
Derive an equation for location of the point 𝑷(𝑢, 𝑣) on the swept surface.
Derive the equation for the swept surface normal 𝑵(𝑢, 𝑣).
 

3. We are given a 2D cubic Bezier curve segment, which has the following control points: 

𝒑𝟎 = (2, 3)

𝒑𝟏 = (3, 0)

𝒑𝟐 = (4, 4)

𝒑𝟑 = (5, 1)

 

Draw the convex hull for this 2D Bezier curve segment.
Compute the value of 𝑃’(0) for this 2D Bezier curve segment.
We are now given a second 2D Bezier curve segment, which has the control points:
𝒑𝟎 = (0, -1)

𝒑𝟏 = (-2, 4)

𝒑𝟐 = (0, 9)

𝒑𝟑 = (2, 3)

Does this segment join the previous segment with C1 continuity? Give a mathematical justification for your answer.

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.
 

We are given the following boundary conditions for a cubic spline section: 
                                                                     𝑷(0) = 𝒑

                                                                       𝑷(1) = 𝒑     

                                                                       𝑷′(0) =     [(1 + 𝑏)(𝒑    − 𝒑       ) + (1 − 𝑏)(𝒑        − 𝒑 )]

                                                                       𝑷′(1) =     [(1 + 𝑏)(𝒑         − 𝒑 ) + (1 − 𝑏)(𝒑         − 𝒑      )]

In the textbook, we see this is a Cardinal Spline (Kochanek-Bartels spline with t=0 and c=0). In this case 𝑀    = [𝑝        𝑝  𝑝      𝑝        ] and the boundary conditions can be written:

                                                   𝑷(0)             0            1           0           0       𝒑

                                                 ⎡𝑷(1)⎤      ⎡    0            0           1           0   ⎤    𝒑

                                                ⎢⎢𝑷′(0)⎥⎥ = ⎢⎢ (       )           𝑏         (       )          0   ⎥⎥ 𝒑      

                                                ⎣𝑷′(1)⎦     ⎣    0           (       )         𝑏         (        )⎦ 𝒑

Show how to compute the 4×4 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.
Given MC write out the blending functions for this curve.
Do adjacent segments satisfy C1 continuity? Give a mathematical justification.
Does adjusting b change the tangent direction at the endpoints or only magnitude?