- Asked 2/25
- 2074 Chaitra · 6+2 marks
Bezier points at u=0.25,0.5,0.75: (2,10)...
Short answer
Using the cubic Bezier formula with control points (2,10),(6,20),(12,5),(16,15): P(0.25)=(5.31,13.59), P(0.5)=(9,12.5), P(0.75)=(12.69,11.41). The curve starts at (2,10), bends toward P1 and P2, ends at (16,15), staying inside the convex hull.
Find the coordinates at u = 0.25, u = 0.5, and u = 0.75 with respect to the control points (2, 10), (6, 20), (12, 5) and (16, 15) using Bezier function. And plot the Bezier curve with your calculated coordinates.
Control points: P0(2,10), P1(6,20), P2(12,5), P3(16,15). Cubic Bezier:
P(u) = (1−u)³P0 + 3u(1−u)²P1 + 3u²(1−u)P2 + u³P3
Blending functions
| u | B0 = (1−u)³ | B1 = 3u(1−u)² | B2 = 3u²(1−u) | B3 = u³ |
|---|---|---|---|---|
| 0.25 | 0.4219 | 0.4219 | 0.1406 | 0.0156 |
| 0.50 | 0.1250 | 0.3750 | 0.3750 | 0.1250 |
| 0.75 | 0.0156 | 0.1406 | 0.4219 | 0.4219 |
u = 0.25 x = 0.4219·2 + 0.4219·6 + 0.1406·12 + 0.0156·16 = 0.844 + 2.531 + 1.688 + 0.250 = 5.3125 y = 0.4219·10 + 0.4219·20 + 0.1406·5 + 0.0156·15 = 4.219 + 8.438 + 0.703 + 0.234 = 13.594
u = 0.5 x = 0.125·2 + 0.375·6 + 0.375·12 + 0.125·16 = 0.25 + 2.25 + 4.5 + 2 = 9 y = 0.125·10 + 0.375·20 + 0.375·5 + 0.125·15 = 1.25 + 7.5 + 1.875 + 1.875 = 12.5
u = 0.75 x = 0.0156·2 + 0.1406·6 + 0.4219·12 + 0.4219·16 = 0.031 + 0.844 + 5.063 + 6.750 = 12.6875 y = 0.0156·10 + 0.1406·20 + 0.4219·5 + 0.4219·15 = 0.156 + 2.813 + 2.109 + 6.328 = 11.406
Answer
| u | x | y |
|---|---|---|
| 0 | 2 | 10 |
| 0.25 | 5.31 | 13.59 |
| 0.5 | 9.00 | 12.50 |
| 0.75 | 12.69 | 11.41 |
| 1 | 16 | 15 |
Plot: mark the control points P0 to P3 (joined by the dashed control polygon), then the calculated points, and draw a smooth curve from P0 through (5.31, 13.59), (9, 12.5) and (12.69, 11.41) to P3.
20 | o P1
|
15 | o P3
| * * *
10 | o P0
|
5 | o P2
+---------------------------------- x
2 6 9 12 16
o = control points, * = calculated points at u = 0.25, 0.5, 0.75
The curve starts at P0, bends towards P1, dips towards P2 and ends at P3, staying inside the convex hull of the four control points.
✓ Answer checked by Claude Opus 5.5 (high effort)
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