Difficulty: Intermediate | Prerequisites: Vectors and dot products (Lab 1A, Part 1). Basic MATLAB plotting commands (plot, linspace).
Parametric curves extend the idea of a function y = f(x) by letting both x and y depend on a third variable, the parameter t. Instead of "y is a function of x," you have "x and y are both functions of t," which means curves can loop, cross themselves, and trace paths that ordinary functions cannot. This is the gateway to describing motion in a plane (position as a function of time) and, shortly after, motion in 3D space. If you understood parameter families of vectors from the first half of Lab 1A, you already have the core idea: each value of t gives you a point.
A parametric curve defines x(t) and y(t) separately, then plots them together to trace a path. You can also plot each component against t on its own to see how that dimension changes. Two parametric curves intersect where they pass through the same (x, y) point, which you find by overlaying their Cartesian plots.
Parametric curve
A curve defined by a pair (or triple) of functions x(t), y(t) (and z(t) in 3D), where t is the parameter. The curve is the set of all points (x(t), y(t)) as t ranges over some interval.
Think of it as: a set of instructions that say "at time t, be at this position."
Component functions
The individual coordinate functions that make up a parametric curve. For C(t) = ⟨x(t), y(t)⟩, the component functions are x(t) and y(t) separately.
In simple terms, this means: each axis gets its own formula.
Cartesian plot
A plot of y versus x on the standard coordinate axes. For a parametric curve, you compute x(t) and y(t) for many values of t and plot the resulting (x, y) points.
Think of it as: the "bird's eye" view of the path, with t hidden.
Intersection (of curves)
A point where two curves cross, meaning they share the same (x, y) coordinates. For parametric curves, the t-values at which each curve reaches the intersection point may differ.
In simple terms, this means: the curves pass through the same spot, but not necessarily at the same "time."
Parameter
The independent variable (usually t) that drives a parametric curve. It is often thought of as time, but it can represent any continuously varying quantity.
Think of it as: the dial that controls where you are along the curve.
In the lab, two curves C1 and C2 are defined by their component functions over a range of t values.
C1 and C2 each produce a set of (x, y) pairs that, when plotted, trace distinct paths in the Cartesian plane.
C1 is plotted in blue, C2 in red, following the lab's convention.
To plot a parametric curve, compute x(t) and y(t) for each sample point of t, then plot y against x.
This produces the curve as it would appear in the xy-plane, with the parameter t not visible on either axis.
In MATLAB: plot(x1, y1, 'b') plots C1 in blue, plot(x2, y2, 'r') plots C2 in red. Use hold on to overlay them.
You can also plot x(t) and y(t) each against t separately. This shows how each coordinate changes as t increases.
The lab's Problem 8 does this: the left graph shows x1 and y1 of C1 plotted against t, and the right graph does the same for C2.
This view is useful for spotting where a component has a maximum, minimum, or changes sign, which is harder to read off the Cartesian curve directly.
Two parametric curves intersect where they pass through the same (x, y) point.
Visually, this is where the blue and red curves cross on the Cartesian plot (Problem 7).
Finding intersections algebraically means solving x₁(t₁) = x₂(t₂) and y₁(t₁) = y₂(t₂) simultaneously. The parameter values t₁ and t₂ need not be the same.
If the curves are at the same point at the same parameter value (t₁ = t₂), that is called a collision, not merely an intersection.
General 2D parametric curve:
C(t) = \langle x(t),\; y(t) \rangle, \quad t \in [a, b]General 3D parametric curve:
C(t) = \langle x(t),\; y(t),\; z(t) \rangle, \quad t \in [a, b]Intersection condition for two curves C₁ and C₂:
x_1(t_1) = x_2(t_2) \quad \text{and} \quad y_1(t_1) = y_2(t_2)Parametric curves describe any motion where position depends on time: the orbit of a planet, the path of a projectile, the trajectory of a robot arm. In computer animation, every moving object follows a parametric path, with t as the frame clock.
Students often think an intersection means both curves arrive at the same point at the same t-value. It does not. An intersection only requires the same (x, y) coordinates; the parameter values can differ. When the parameter values also match, it is a collision.
Plotting x versus t is not the same as plotting the parametric curve. The parametric (Cartesian) plot shows y versus x with t eliminated. The component plot shows one coordinate versus t.
Students sometimes assume a parametric curve must pass the vertical line test. It does not. Parametric curves can loop, cross, and double back, because they are not functions of x.
Confusing the direction of traversal with the shape of the curve. Two parametric curves can trace the same shape but in opposite directions (or at different speeds) depending on how t is mapped.
⚠️ Be able to sketch a parametric curve from its component functions, and vice versa. This means converting between the (x vs t, y vs t) view and the (y vs x) Cartesian view.
⚠️ Know the difference between an intersection and a collision. Exam questions may give two curves and ask whether they meet, and if so, whether they collide.
⚠️ Understand that parametric curves are not restricted to passing the vertical line test. Circles, figure-eights, and spirals are all fair game.
⚠️ Expect MATLAB syntax questions: how to define t, how to plot, how to overlay two curves with hold on.
True or false: A parametric curve in 2D is defined by two functions, x(t) and y(t).
Fill in the blank: Plotting y(t) versus x(t) with t hidden produces a ____ plot.
True or false: If two parametric curves intersect, they must reach the intersection point at the same t-value.
True or false: A parametric curve can cross itself.
Fill in the blank: Plotting x(t) versus t on its own shows how the ____ component changes with the parameter.
Answers: 1. True. 2. Cartesian. 3. False (only collisions require the same t). 4. True. 5. x (horizontal).
Q: Given C(t) = ⟨t², sin(t)⟩ for t ∈ [0, 2π], what does the Cartesian plot show?
A: A curve where the x-coordinate grows as t² (always positive, increasing) and the y-coordinate oscillates between -1 and 1. The result is a wave-like path that moves to the right.
Q: How do you determine whether two parametric curves intersect?
A: Plot both curves in the same Cartesian coordinate system and look for crossing points. Algebraically, solve x₁(t₁) = x₂(t₂) and y₁(t₁) = y₂(t₂) simultaneously.
Q: What is the difference between plotting y versus x and plotting x versus t?
A: The y-versus-x plot shows the shape of the path in the plane. The x-versus-t plot shows how the horizontal position changes over the parameter, which reveals speed and direction information the Cartesian plot hides.
Q: Can a parametric curve fail the vertical line test? Why or why not?
A: Yes. A parametric curve is not a function of x, so it can have multiple y-values for the same x. Circles, loops, and figure-eights are all valid parametric curves.
Q: If two curves cross at the same (x, y) but at different parameter values, is that an intersection or a collision?
A: An intersection. A collision requires the curves to be at the same point at the same parameter value.
Parametric curves connect directly to vector-valued functions (the first half of this lab), because a parametric curve is simply the path traced by a vector-valued function's tip. This leads to arc length (integrating the norm of the derivative), curvature (how sharply the curve bends), and eventually line integrals and Green's theorem. The idea of decomposing motion into components (x vs t, y vs t) is the same decomposition used in physics for projectile motion.
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