---
title: "Parametric cubic curves"
subject: "Computer Graphics"
chapter: 5
key: "semester_5/computer_graphics:parametric-cubic-curve"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/parametric-cubic-curve
site: BCT Engineering
---

# Parametric cubic curves

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2073 Shrawan, 2 marks; 2072 Chaitra, 2 marks; 2073 Chaitra, 2 marks; 2068 Chaitra, 3 marks; 2078 Bhadra, 3 marks; 2078 Kartik, 4 marks (asked 6 times)
- Video: [Parametric Curve in Computer Graphics Explained in Hindi](https://www.youtube.com/watch?v=5PMkHqlaP88) — 5 Minutes Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/parametric-cubic-curve
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

What is a parametric cubic curve and why do you need it (is it necessary to construct curves using parametric equations? Justify)?

## Short answer

A parametric cubic curve defines coordinates as cubic polynomials in parameter u. It's needed because it handles multivalued curves, infinite slopes, is coordinate-independent, and allows easy tangent calculations and smooth joins. Cubic degree provides control over both endpoints and tangents without excessive oscillations.

## Answer

A **parametric cubic curve** is a curve whose x, y, z coordinates are each a cubic polynomial in a parameter u (0 ≤ u ≤ 1):

x(u) = a_x u³ + b_x u² + c_x u + d_x
y(u) = a_y u³ + b_y u² + c_y u + d_y
z(u) = a_z u³ + b_z u² + c_z u + d_z

In matrix form: P(u) = [u³ u² u 1] · C, where C is the 4×3 matrix of coefficients, found from end points and tangents (Hermite) or control points (Bezier).

**Why parametric equations are needed**
1. A curve may be multivalued (a circle has two y values per x), but each coordinate is a single-valued function of u.
2. A vertical tangent has infinite slope in y = f(x) form, but is handled easily as dx/du = 0.
3. They are independent of the coordinate system; they work equally for 2D and 3D curves.
4. Tangent vectors are simply P'(u) = [3u² 2u 1 0]·C.
5. Curves can be joined smoothly by matching end points and derivatives.

**Why cubic?** Lower degrees (linear, quadratic) cannot control both end points and end tangents, and cannot have an inflection. Higher degrees need more computation and cause unwanted wiggles. Cubic has four coefficients per coordinate, matching two end points plus two end tangents, and gives C² smooth joins.

## Check yourself

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

1. A parametric cubic curve has:
   A. Each coordinate as a cubic polynomial in u
   B. y as a cubic in x only
   C. A straight line in u
   D. Each coordinate linear in u
   Answer: A. x(u), y(u), z(u) are cubics with 0 ≤ u ≤ 1. (Key point: Definition)

2. A circle needs parametric form because:
   A. It is a straight line
   B. It is multivalued in y = f(x) form
   C. It has no tangent
   D. It is 3D
   Answer: B. Each coordinate is single-valued in u. (Key point: Why parametric)

3. A vertical tangent in parametric form is handled as:
   A. Infinite slope error
   B. dy/du = 0
   C. dx/du = 0
   D. Undefined
   Answer: C. No infinite slope appears. (Key point: Vertical tangent)

4. Cubic is chosen because:
   A. It is the lowest degree
   B. Quadratics have inflections
   C. It needs no computation
   D. Four coefficients match two end points and two end tangents
   Answer: D. Lower degrees cannot control both end tangents; higher degrees wiggle. (Key point: Why cubic)

## Related in Computer Graphics

- [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
- [Hermite curve and blending functions](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ahermite-curve-blending-functions) — Computer Graphics, chapter 5, asked 2×
- [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

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- [Parametric and non-parametric curve representation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Aparametric-vs-nonparametric-curves) — Computer Aided Design, chapter 2
- [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
- [Need for load flow; NR and decoupled equations](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aload-flow-need-nr-decoupled) — Power System Analysis II, chapter 2