Given below are two statements: Statement 𝐼: Bezier curves are curves that…
2020
Given below are two statements:
Statement 𝐼: Bezier curves are curves that interpolate all of their control points
Statement 𝐼𝐼: A cubic bezier curve has four control points
In the light of the above statements, choose the correct answer from the options given below
- A.
Both Statement 𝐼 and Statement 𝐼𝐼 are true
- B.
Both Statement 𝐼 and Statement 𝐼𝐼 are false
- C.
Statement 𝐼 is correct but Statement 𝐼𝐼 is false
- D.
Statement 𝐼 is incorrect but Statement 𝐼𝐼 is true
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Correct answer: D
Answer: Statement I is incorrect; Statement II is true.
Why Statement I is incorrect:
A Bézier curve is defined by control points that influence the shape of the curve. The curve generally passes through the first and last control points (endpoints), but intermediate control points do not usually lie on the curve; they pull the curve toward them rather than being interpolated.
Why Statement II is true:
A cubic Bézier curve is specified by four control points, commonly labeled P0, P1, P2, and P3.
Its parametric form (for 0 ≤ t ≤ 1) is B(t) = (1 - t)^3 P0 + 3t(1 - t)^2 P1 + 3t^2(1 - t) P2 + t^3 P3, which uses exactly four control points.
Conclusion: Statement I is false because Bézier curves do not interpolate all control points (only endpoints are guaranteed); Statement II is true because a cubic Bézier curve is defined by four control points.