Computer Graphics » Blog Archive

Week 4 Tutorial

Remember, you must attempt the tutorial problems before the tutorial.

Question 1

The four control points of a Bezier curve B(t) are (0,0), (0,80), (80,80) and (80,0). What is B(0.5)? Sketch the curve as accurately as you can. Show the tangents to the curve at t=0 and t=1.

If this Bezier is used as the centre line for a road of width 20 as in assigment one, where will the points on the edge of the road corresponding to B(1/2) end up?

Question 2

Show that we can transform (rotate, scale, or translate) a Bezier curve by transforming the control points.

Question 3

Curve

is a Bezier curve with control points P_0,P_1,P_2,P_3

. Curve

is a Bezier curve with control points P_3,P_4,P_5,P_6

. What are the conditions on P_3,P_4,P_5,P_6

for the curves to meet with C^2

continuity? (Hint: the curves meet with C^2

continuity if A''(1)=B''(0)

Question 4

De Casteljau’s algorithm calculates:

$\begin{array}{rcl} P_{01} &=& (P_0+P_1)/2\\ P_{12} &=& (P_1+P_2)/2\\ P_{23} &=& (P_2+P_3)/2\\ P_{012} &=& (P_{01}+P_{12})/2\\ P_{123} &=& (P_{12}+P_{23})/2\\ P_{0123} &=& (P_{012}+P_{123})/2\end{array}$

Show that $P_{0123}$ is the mid point of the Bezier curve with control points:

Question 5

The basis functions for a uniform B spline are

$\begin{array}{rcl}b_0(t) &=& (-t^3 + 3t^2 -3t +1)/6\\ b_1(t) &=& (3t^3 - 6t^2 +4)/6\\ b_2(t) &=& (-3t^3 + 3t^2 +3t +1)/6\\ b_3(t) &=& t^3/6\end{array}$

and the B spline is

$B_i(t)=\sum_{i=0}^3 b_i(t)P_{i+j}$

(a) Show that the uniform B spline has the convex hull property.

(b) Show that the B spline has C^1 continuity. That is, show:

This entry was posted on Thursday, August 14th, 2008 at 8:05 pm and is filed under Tutorial Questions. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

Comments are closed.