---
title: "Convert Hermite data to Bezier cubic"
subject: "Computer Aided Design"
chapter: 3
key: "semester_4/computer_aided_design:hermite-to-bezier-conversion"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/hermite-to-bezier-conversion
site: BCT Engineering
---

# Convert Hermite data to Bezier cubic

Computer Aided Design · Chapter 3: Mathematical Representation of Curves

- Asked in: year not given, 6 marks
- Video: [Splines in 5 minutes:  Part 1 -- cubic curves](https://www.youtube.com/watch?v=YMl25iCCRew) — Graphics in 5 Minutes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/hermite-to-bezier-conversion
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-3

## Question

A Hermite cubic has P0 = (0, 0), P1 = (6, 0), P0' = (3, 6) and P1' = (3, -6). Derive the relation between Hermite and cubic Bezier data and find the Bezier control points. Hence calculate the curve point at u = 0.5 and u = 0.25.

## Short answer

The Bezier control points are B0 = P0 = (0,0), B1 = P0 + P0'/3 = (1,2), B2 = P1 - P1'/3 = (5,2), B3 = P1 = (6,0). Using Bernstein blending, P(0.5) = (3.0, 1.5) and P(0.25) = (1.219, 1.125).

## Answer

#### Relation between the forms
A cubic Bezier curve with control points $\mathbf B_0..\mathbf B_3$ has end points $\mathbf P(0)=\mathbf B_0$, $\mathbf P(1)=\mathbf B_3$ and end tangents

$$
\mathbf P'(0)=3(\mathbf B_1-\mathbf B_0),\qquad \mathbf P'(1)=3(\mathbf B_3-\mathbf B_2)
$$

If this curve is to equal the Hermite curve, the end conditions must be identical:

$$
\mathbf B_0=\mathbf P_0,\quad \mathbf B_3=\mathbf P_1,\quad
\mathbf B_1=\mathbf P_0+\tfrac13\mathbf P_0',\quad
\mathbf B_2=\mathbf P_1-\tfrac13\mathbf P_1'
$$

Both are cubics with the same four end conditions, so they are the same curve.

#### Control points
$$
\begin{aligned}
\mathbf B_0&=(0,0) \\
\mathbf B_1&=(0,0)+\tfrac13(3,6)=(1,\ 2) \\
\mathbf B_2&=(6,0)-\tfrac13(3,-6)=(5,\ 2) \\
\mathbf B_3&=(6,0)
\end{aligned}
$$

#### Point at u = 0.5
Weights $0.125, 0.375, 0.375, 0.125$:

$$
x=0.375(1)+0.375(5)+0.125(6)=3.0,\qquad y=0.375(2)+0.375(2)=1.5
$$

#### Point at u = 0.25
Weights $0.421875,\ 0.421875,\ 0.140625,\ 0.015625$:

$$
\begin{aligned}
x&=0.421875(1)+0.140625(5)+0.015625(6)=1.2188 \\
y&=0.421875(2)+0.140625(2)=1.125
\end{aligned}
$$

**Answer: Bezier control points (0,0), (1,2), (5,2), (6,0); P(0.5) = (3.0, 1.5); P(0.25) = (1.219, 1.125).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is B1 in terms of Hermite data?
   A. P0 + P0'
   B. P0 - P0'/3
   C. P0 + P0'/3
   D. P1 + P1'/3
   Answer: C. Since P'(0) = 3(B1 - B0), B1 = P0 + P0'/3. (Key point: Relation)

2. What is B2?
   A. P1 + P1'/3
   B. P1 - P1'
   C. P0 + P1'/3
   D. P1 - P1'/3
   Answer: D. P'(1) = 3(B3 - B2), so B2 = P1 - P1'/3. (Key point: B2)

3. What are the Bezier control points?
   A. (0,0),(1,2),(5,2),(6,0)
   B. (0,0),(3,6),(3,-6),(6,0)
   C. (0,0),(1,2),(5,-2),(6,0)
   D. (0,0),(3,2),(3,2),(6,0)
   Answer: A. B1 = (1,2), B2 = (6,0) - (1,-2) = (5,2). (Key point: Control points)

4. What is P(0.5)?
   A. (3.0, 2.0)
   B. (3.0, 1.5)
   C. (2.5, 1.5)
   D. (3, 0)
   Answer: B. x = 0.375 + 1.875 + 0.75 = 3, y = 0.75 + 0.75 = 1.5. (Key point: Midpoint)

## Related in Computer Aided Design

- [Derive Hermite cubic curve and blending functions](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Ahermite-cubic-derivation) — Computer Aided Design, chapter 3
- [Hermite curve points and tangents](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Ahermite-curve-numerical) — Computer Aided Design, chapter 3
- [Cubic Bezier point and tangent calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-curve-numerical) — Computer Aided Design, chapter 3
- [Uniform cubic B-spline segment points](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Auniform-cubic-bspline-numerical) — Computer Aided Design, chapter 3
- [Biquadratic Bezier surface point calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-surface-numerical) — Computer Aided Design, chapter 4
- [B-spline curve versus Bezier curve](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abspline-vs-bezier) — Computer Aided Design, chapter 3

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