To parametrize a curve, you express each coordinate (x, y, and sometimes z) as a separate function of a single variable, usually called t. Instead of describing a curve with one equation like y = x², you write x(t) and y(t) so that as t changes, the point traces out the curve. Think of t as time: at each moment, the functions tell you exactly where a point sits on the curve.
What Parametrization Actually Means
A parametrization is a map r(t) = ⟨x(t), y(t)⟩ from a parameter interval [a, b] to the plane. The functions x(t) and y(t) are called coordinate functions. In three dimensions, you add a third: r(t) = ⟨x(t), y(t), z(t)⟩. As t moves from a to b, the endpoint of this vector traces out the curve, just like a pen drawing a path across a page.
The parameter t doesn’t have to represent time, but thinking of it that way helps. At t = 0, the point is somewhere specific. At t = 1, it’s moved. The parametrization captures not just the shape of the curve but the direction and speed at which a point moves along it.
The Simplest Case: Graphs of Functions
If you already have a function y = f(x), converting it to parametric form is straightforward. Set x = t, then y = f(t). For example, the parabola y = x² becomes x(t) = t, y(t) = t². The graph is identical. You’re just rewriting the same relationship using a parameter. This works for any function where y depends on x, and it’s often the fastest way to get started with parametric equations.
Parametrizing a Line Segment
To parametrize the line segment from point A to point B, use the formula:
r(t) = (1 − t)A + tB, for 0 ≤ t ≤ 1.
At t = 0 you’re at A, and at t = 1 you’re at B. Every value of t between 0 and 1 gives a point along the segment. For a concrete example, the segment from (1, 3) to (4, 7) becomes x(t) = 1 + 3t and y(t) = 3 + 4t, where each coordinate moves linearly from its starting value to its ending value. If you need the segment to run over a different interval of t, you can rescale, but [0, 1] is the standard choice.
Circles and Ellipses
Circles and ellipses have natural parametrizations using sine and cosine. For a circle centered at (h, k) with radius r:
- x(t) = r cos(t) + h
- y(t) = r sin(t) + k
As t runs from 0 to 2π, the point completes one full loop. For an ellipse with semi-axes a (horizontal) and b (vertical) centered at (h, k), you replace the single radius with two:
- x(t) = a cos(t) + h
- y(t) = b sin(t) + k
This works because cos²(t) + sin²(t) = 1, which is exactly the algebraic identity behind circles and ellipses. If you want an object to trace the path faster, multiply t by a constant inside the trig functions. Using cos(2t) and sin(2t), for instance, completes the loop in π units of time instead of 2π.
3D Curves and Helices
In three dimensions, you add a z(t) component. The classic example is a helix: r(t) = ⟨cos(t), sin(t), t⟩. The x and y coordinates trace a circle while z climbs steadily, producing a spiral staircase shape. The radius of the helix is controlled by the coefficient in front of cos and sin. Writing r(t) = ⟨0.5 cos(t), 0.5 sin(t), t⟩ gives a tighter helix with radius 0.5 instead of 1. The rate of climb is controlled by the coefficient on t in the z-component.
Direction and Orientation
A parametrization doesn’t just describe where a curve is. It also describes which direction you travel along it. As t increases, the point moves in a specific direction, and this is called the orientation of the curve. For the circle x(t) = cos(t), y(t) = sin(t), increasing t traces the circle counterclockwise. If you want clockwise motion, swap to x(t) = cos(t), y(t) = −sin(t).
More generally, if you replace t with −t in a parametrization, you reverse the direction of travel. The geometric curve is the same, but the orientation flips. This matters in calculus when computing line integrals, where the direction of travel affects the sign of the result. A key property of smooth parametrizations is that the velocity vector never reaches zero, meaning the point never stops and reverses. If it did, the orientation would be ambiguous.
Piecewise Curves
Sometimes you need to parametrize a path made up of several distinct pieces, like a triangle or a path that follows a line and then curves. The strategy is to parametrize each piece separately, then shift the parameter intervals so they connect end to end.
Say the first piece naturally uses t from 0 to 3, and the second piece naturally uses t from −4 to −1. You want the second piece to pick up where the first left off at t = 3. Add 7 to the parameter of the second piece so its interval becomes 3 to 6. Then substitute back so the formulas reflect the shift. If a third piece follows, shift its interval to start at 6, and so on. The result is a single parameter t that runs continuously through all pieces, with each segment defined on its own sub-interval.
Arc Length Parametrization
Most parametrizations move through the curve at varying speeds. In a circle parametrized with cos(t) and sin(t), the speed happens to be constant, but that’s a special case. An arc length parametrization is one where the parameter directly measures distance traveled along the curve. One unit of parameter change always corresponds to one unit of distance.
To find it, you first compute the arc length function: s(t) equals the integral of the speed (the magnitude of the velocity vector) from some starting point to t. Because speed is always positive for a smooth curve, s(t) is strictly increasing, which means you can solve for t in terms of s. Plugging that inverse back into the original parametrization gives you the arc length version. In practice, this integral is often difficult to evaluate, so arc length parametrizations are more of a theoretical tool than something you compute by hand for every curve. But they’re important in differential geometry because they strip away the arbitrary choice of speed and isolate the pure shape of the curve.
Velocity and Acceleration From Parametric Equations
Once a curve is parametrized, calculus gives you powerful tools. The first derivative r'(t) is the velocity vector, pointing in the direction of motion with a magnitude equal to speed. The second derivative r”(t) is the acceleration vector. These let you analyze motion along the curve in physical terms.
Acceleration splits into two components. The tangential component measures how the speed is changing: are you speeding up or slowing down? The normal component measures how sharply the path is turning. For an object moving at constant speed along a curve, the tangential acceleration is zero, and all the acceleration points inward, perpendicular to the direction of travel. This is exactly what happens in uniform circular motion, where the acceleration always points toward the center of the circle.
Choosing the Right Approach
The method you use depends on the shape of the curve:
- Function graphs (y = f(x)): set x = t, y = f(t)
- Line segments: use r(t) = (1 − t)A + tB on [0, 1]
- Circles: use cos(t) and sin(t) scaled by the radius
- Ellipses: use a cos(t) and b sin(t) for the two semi-axes
- Helices: circular x and y with a linear z component
- Piecewise paths: parametrize each segment, then shift intervals to connect them
The same curve can have infinitely many valid parametrizations. You might use t from 0 to 1, or from 0 to 2π, or move through the curve at different speeds. All of these describe the same geometric shape. The “best” parametrization is the one that makes your particular calculation easiest, whether that’s evaluating an integral, computing a tangent vector, or simulating motion along a path.