Parametric and Bézier Curves using Vectors

Problem Statement

When a particle moves through space, its $x$ and $y$ coordinates often change independently over time. We can model this by defining both $x$ and $y$ in terms of a third variable, or parameter, usually time $t$ or an angle $\theta$.

Vectors provide the perfect container for these parametric equations: $\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}$. While parametric forms are excellent for calculating velocity and acceleration, we often want to know the physical shape of the path the particle took. To do this, we must eliminate the parameter to find the Cartesian equation (an equation involving only $x$ and $y$).

Furthermore, this concept of parametric vector generation is the foundation of Bézier curves—the mathematical curves used in computer graphics, game design, and font rendering to draw smooth, scalable shapes.

Consider a particle whose position vector at time $t$ is given by: $\mathbf{r}(t) = (t^2 + 1)\mathbf{i} + (t - 2)\mathbf{j}$ for all real values of $t$.

(a) Write down the parametric equations for $x$ and $y$.

(b) Eliminate the parameter $t$ to find the Cartesian equation of the curve traced by the particle.

(c) Describe the geometric shape of this Cartesian curve and determine its vertex.

Hints

• Part (a): Simply extract the components of the $\mathbf{i}$ and $\mathbf{j}$ vectors.
• Part (b): Look at your two parametric equations. One of them is linear (degree 1). Rearrange this linear equation to make $t$ the subject. Then, substitute this expression for $t$ into the other, non-linear equation. Expand and simplify to get an equation with only $x$ and $y$.
• Part (c): Look at the final Cartesian equation. Is it linear? Is it a circle ($x^2 + y^2 = r^2$)? Is it a parabola ($y = x^2$ or $x = y^2$)? Compare it to standard conic forms to find the vertex.

Solutions

Part (a): Parametric Equations

1. From the position vector $\mathbf{r}(t) = (t^2 + 1)\mathbf{i} + (t - 2)\mathbf{j}$, we extract the components:
   • $x = t^2 + 1$
   • $y = t - 2$

Part (b): Eliminating the Parameter

1. We want an equation linking $x$ and $y$ directly. We use the linear equation $y = t - 2$ to isolate $t$: $t = y + 2$
2. Substitute this expression for $t$ into the $x$ equation: $x = (y + 2)^2 + 1$
3. Expand the perfect square: $x = (y^2 + 4y + 4) + 1$ $x = y^2 + 4y + 5$
4. This is the Cartesian equation of the curve. Alternatively, leaving it in vertex form $x = (y + 2)^2 + 1$ is often more useful for graphing.

Part (c): Identifying the Curve

1. The equation $x = y^2 + 4y + 5$ features a $y^2$ term and an $x$ term. This indicates that it is a horizontal parabola (opening to the right, since the coefficient of $y^2$ is positive).
2. To find the vertex, we look at the completed square form: $x = (y + 2)^2 + 1$.
3. Standard horizontal parabola vertex form is $(x - h) = a(y - k)^2$, where the vertex is $(h, k)$.
4. Comparing the two: $(x - 1) = 1(y - (-2))^2$.
5. Therefore, the vertex of the parabola is $(1, -2)$.

Takeaways

• The Power of Parameters: Vector equations naturally encapsulate parametric paths. They tell you not just where a curve is, but when a particle is at a specific point on that curve.
• Elimination Strategy: Always look for the simplest parametric equation (usually the linear one, or one involving basic trig identities like $\sin^2\theta + \cos^2\theta = 1$) to eliminate the parameter.
• Real-World Application: The curves drawn in Adobe Illustrator, the smooth edges of the fonts you are reading right now, and the paths of animated characters in video games are all constructed using advanced parametric vector equations known as Bézier curves.

Further Readings

• HSC Vectors: https://vumaths.com/booklets/hsc-vectors/
• HSC Polynomials: https://vumaths.com/booklets/hsc-polynomials/
• HSC Functions: https://vumaths.com/booklets/hsc-functions/

Connect with me

Mastering parametric conversions is essential for both Extension 1 and Extension 2 HSC Mathematics. If you need more practice, check out the booklets on Vu's Maths Hub. You can also watch my video explanations on my YouTube channel. Follow my Instagram for bite-sized tips, and read my extended thoughts on the syllabus on Substack. Let's connect!