---
title: "Parametric and non-parametric curve representation"
subject: "Computer Aided Design"
chapter: 2
key: "semester_4/computer_aided_design:parametric-vs-nonparametric-curves"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/parametric-vs-nonparametric-curves
site: BCT Engineering
---

# Parametric and non-parametric curve representation

Computer Aided Design · Chapter 2: Geometric Modeling Fundamentals

- Asked in: year not given, 6 marks
- Video: [Parametric & Non Parametric Representation of Curve - Technique For Geometric Modeling - CAD/CAM/CAE](https://www.youtube.com/watch?v=3gbxDyivUx0) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/parametric-vs-nonparametric-curves
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-2

## Question

Differentiate between parametric and non-parametric representation of curves. Why is parametric representation preferred in CAD? Represent a circle of radius 5 centred at the origin in both forms.

## Short answer

Non-parametric form is explicit y=f(x) or implicit f(x,y)=0; parametric form is P(u)=[x(u),y(u),z(u)]. Parametric is preferred because it handles multi-valued and closed curves, vertical tangents, bounded segments and easier transformations. Circle radius 5: x^2+y^2=25; parametric x=5 cos(theta), y=5 sin(theta), 0<=theta<=2pi.

## Answer

**Non-parametric** representation gives a direct relation between coordinates: explicit $y=f(x)$ or implicit $f(x,y)=0$. **Parametric** representation expresses each coordinate as a function of an independent parameter $u$: $x=x(u)$, $y=y(u)$, $z=z(u)$, with $u$ in a range, usually $0\le u\le1$.

| Point | Non-parametric | Parametric |
|---|---|---|
| Form | $y=f(x)$ or $f(x,y)=0$ | $\mathbf P(u)=[x(u),y(u),z(u)]$ |
| Multi-valued curves (circle) | Explicit form fails (two $y$ values per $x$) | No problem |
| Vertical tangent | Slope becomes infinite | $dy/du$, $dx/du$ remain finite |
| Axis dependence | Depends on chosen axes | Independent of axes |
| Point generation | Solve equation, may need iteration | Substitute $u$ directly |
| Bounded segments | Needs extra conditions | Limits of $u$ define the segment |
| 3D curves | Needs two equations | Easy to extend |
| Transformation | Equation must be re-derived | Transform control points / vector |

#### Why parametric is preferred
1. Curves can be closed, multi-valued or have vertical tangents.
2. Easy to generate points at equal steps of $u$ for display and tool paths.
3. Tangent vector is simply $d\mathbf P/du$.
4. Segments are bounded and can be joined with continuity conditions.
5. Same equation works in 2D and 3D, and with any axes.

#### Circle of radius 5
Non-parametric (implicit): $x^2+y^2=25$; explicit form needs two branches $y=\pm\sqrt{25-x^2}$.

Parametric:

$$
x = 5\cos\theta,\qquad y = 5\sin\theta,\qquad 0\le\theta\le 2\pi
$$

For example, $\theta=0$ gives $(5,0)$ directly, where the explicit form has an infinite slope ($dy/dx\to\infty$), while the parametric tangent $(dx/d\theta,\ dy/d\theta)=(0,\ 5)$ stays finite.

## Check yourself

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

1. Why does the explicit form y = f(x) fail for a circle?
   A. Each x gives two y values
   B. It has no equation
   C. It cannot be drawn in 2D
   D. Slope is always zero
   Answer: A. A circle is multi-valued: y = ±sqrt(25 - x^2) gives two branches. (Key point: Circle explicit form)

2. How do parametric forms handle a vertical tangent?
   A. Slope becomes infinite
   B. dx/du and dy/du remain finite
   C. They cannot
   D. They use a different axis
   Answer: B. The tangent vector (dx/du, dy/du) stays finite, unlike the slope dy/dx. (Key point: Vertical tangent)

3. Which is the parametric form of a circle of radius 5 at the origin?
   A. x = 5 sin θ, y = 5 sin θ
   B. x^2 + y^2 = 5
   C. x = 5 cos θ, y = 5 sin θ, 0 ≤ θ ≤ 2π
   D. x = cos θ, y = sin θ
   Answer: C. x = 5 cos θ, y = 5 sin θ for 0 to 2π; the implicit form is x^2 + y^2 = 25. (Key point: Circle parametric)

4. Which is a reason for preferring parametric representation?
   A. It needs fewer equations in 3D
   B. Bounded segments need extra conditions
   C. It requires solving the equation for points
   D. It is independent of axes and tangent is simply dP/du
   Answer: D. Parametric curves are axis-independent, easy to generate point by point, with tangent dP/du. (Key point: Why preferred)

## Related in Computer Aided Design

- [Parametric circle and ellipse with points](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Acircle-ellipse-parametric) — Computer Aided Design, chapter 3

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