---
title: "Hermite curve and blending functions"
subject: "Computer Graphics"
chapter: 5
key: "semester_5/computer_graphics:hermite-curve-blending-functions"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/hermite-curve-blending-functions
site: BCT Engineering
---

# Hermite curve and blending functions

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2081 Chaitra (new course), 4 marks; 2070 Chaitra, 2+6 marks (asked 2 times)
- Video: [Hermite Curve Algorithm in Computer Graphics by BhanuPriya | Lec-36](https://www.youtube.com/watch?v=jqbr39dhyWQ) — AllAboutCSIT
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/hermite-curve-blending-functions
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

Explain Hermite interpolation (Hermite curve for 3-degree polynomial) with the required expressions, i.e. the blending functions of a parametric cubic curve.

## Short answer

A Hermite curve is a cubic interpolating curve defined by endpoints P0, P1 and tangent vectors Dp0, Dp1. Using blending functions: P(u) = P0(2u^3-3u^2+1) + P1(-2u^3+3u^2) + Dp0(u^3-2u^2+u) + Dp1(u^3-u^2). Segments join with C1 continuity by sharing tangents at joints.

## Answer

A **Hermite curve** is a cubic interpolating curve specified by its two end points P0 and P1 and the tangent vectors (derivatives) at those end points, Dp0 and Dp1.

**Derivation:** P(u) = a u³ + b u² + c u + d, so P'(u) = 3a u² + 2b u + c, 0 ≤ u ≤ 1.

Boundary conditions:
- P(0) = d = P0
- P(1) = a + b + c + d = P1
- P'(0) = c = Dp0
- P'(1) = 3a + 2b + c = Dp1

Solving: a = 2P0 − 2P1 + Dp0 + Dp1, b = −3P0 + 3P1 − 2Dp0 − Dp1, c = Dp0, d = P0.

**Matrix form**

```text
                  |  2 -2  1  1 | | P0  |
P(u) = [u³ u² u 1]| -3  3 -2 -1 | | P1  |
                  |  0  0  1  0 | | Dp0 |
                  |  1  0  0  0 | | Dp1 |
```

**Blending (Hermite) functions**

P(u) = P0·H0(u) + P1·H1(u) + Dp0·H2(u) + Dp1·H3(u)

- H0(u) = 2u³ − 3u² + 1
- H1(u) = −2u³ + 3u²
- H2(u) = u³ − 2u² + u
- H3(u) = u³ − u²

H0 + H1 = 1. At u = 0, H0 = 1 and the rest are 0, so P(0) = P0. At u = 1, H1 = 1 so P(1) = P1. Changing the tangent magnitudes changes how tightly the curve bends. Hermite segments are joined with C¹ continuity by using the same tangent at the join.

## Check yourself

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

1. A Hermite cubic curve is specified by:
   A. Four control points on the curve
   B. Two end points and two off-curve control points
   C. Only the end points and the curve degree
   D. Two end points and the tangent vectors at those end points
   Answer: D. Hermite uses P0, P1 and tangents Dp0, Dp1 as its four constraints. (Key point: Hermite specification)

2. Which condition gives the coefficient c in P(u)=au^3+bu^2+cu+d?
   A. P(0) = c
   B. P'(0) = c = Dp0
   C. P'(1) = c
   D. P(1) = c = P1
   Answer: B. Differentiating gives P'(0)=c, so c equals the starting tangent Dp0, while P(0)=d=P0. (Key point: Boundary conditions)

3. What are the values of H0 and H1 at u = 0 and u = 1?
   A. H0(0)=1 and H1(1)=1, so P(0)=P0 and P(1)=P1
   B. H0(0)=0 and H1(1)=0
   C. H1(0)=1 and H0(1)=1
   D. All four functions equal 1 at both ends
   Answer: A. H0=2u^3-3u^2+1 equals 1 at u=0 and H1=-2u^3+3u^2 equals 1 at u=1, so the curve starts at P0 and ends at P1. (Key point: Blending functions)

4. How are two Hermite segments joined with C1 continuity?
   A. Use the same curvature at the join
   B. Use the same tangent vector at the join
   C. Make the end points differ slightly
   D. Use tangents in opposite directions
   Answer: B. C1 means equal first derivatives at the join, i.e. the same tangent. (Key point: Continuity)

## Related in Computer Graphics

- [Parametric cubic curves](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aparametric-cubic-curve) — Computer Graphics, chapter 5, asked 6×
- [Interpolation versus approximation spline](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ainterpolation-vs-approximation-spline) — Computer Graphics, chapter 5, asked 2×
- [Derive cubic Bezier equation and properties](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-cubic-equation-properties) — Computer Graphics, chapter 5
- [Curve modeling using splines](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acurve-modeling-using-splines) — Computer Graphics, chapter 5, asked 3×

## Related in other subjects

- [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
- [Iso/super/sub-parametric; Hermite shape function](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Ahermite-beam-shape-isoparametric) — Computational Techniques in Civil Engineering, chapter 4
- [Convert Hermite data to Bezier cubic](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Ahermite-to-bezier-conversion) — 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