Suppose a Bezier curve P(t) is defined by the following four control points in…
2021
Suppose a Bezier curve P(t) is defined by the following four control points in the xy - plane : \(P_0 = (-2,0); P_1 = (-2,4); P_2 = (2,4); \) and \(P_3 = (2,0)\). Then which of the following statements are correct ?
A. Bezier curve P(t) has degree 3.
B. \(P(\frac 1 2) = (0,3)\).
C. Bezier curve P(t) may extend outside the convex hull of its control points.
Choose the correct answer from the options given below :
- A.
A and B only
- B.
A and C only
- C.
B and C only
- D.
A, B, and C
Attempted by 69 students.
Show answer & explanation
Correct answer: A
Final answer: A and B only.
Reasoning:
Degree of the curve: A Bezier curve defined by four control points is a cubic, i.e. degree 3, because degree = number of control points − 1.
Value at t = 1/2: Use the cubic Bezier formula P(t) = (1−t)^3 P0 + 3t(1−t)^2 P1 + 3t^2(1−t) P2 + t^3 P3. For t = 1/2 the Bernstein weights are 1/8, 3/8, 3/8, 1/8.
Compute x-coordinate: (−2)(1/8) + (−2)(3/8) + (2)(3/8) + (2)(1/8) = 0.
Compute y-coordinate: (0)(1/8) + (4)(3/8) + (4)(3/8) + (0)(1/8) = 3.
Thus P(1/2) = (0, 3).
Convex hull property: Every point on a Bezier curve is a convex combination of the control points because Bernstein basis functions are nonnegative for t in [0,1] and sum to 1. Therefore the curve cannot extend outside the convex hull of its control points, so the statement that it may extend outside is false.
Conclusion: The correct statements are the degree being 3 and P(1/2) = (0,3); the convex hull claim is false.