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Math 133 Parametric Curves Stewart §10.
Back to pictures! We have emphasized four conceptual levels, or points of view on mathematics: physical, geometric, numerical, algebraic. The physical viewpoint is that of Applied Mathematics, including engineering and the hard sciences: useful, powerful, revealing. The numerical point of view, officially called Analysis, is concerned with approximations, error-control, and convergence of limits. It prevents our reasoning from falling into chaos when we deal with infinite shapes or processes, but a liking for Analysis is a special taste even among mathematicians. Algebra, my favorite, is concerned with formulas to consisely construct and transform complicated quantities by means of symbolic operations, often giving amazingly simple answers. But deep down, what we really love in math is Geometry: pictures! In this section, we will learn to handle the simplest geometric objects: curves. So far, we have dealt with curves as graphs of functions y = f (x), in which we imagine the independent variable x moving along its axis while f (x) controls the height.
Parametric lines. A more general model for a curve is to consider it as the path of a particle moving in the plane in any fashion. We specify its coordinates as functions of time: that is, at time t, the particle is at the position (x(t), y(t)). We call the variable t the parameter, and the trajectory traced out is a parametric curve. Any graph y = f (x) can immediately be written parametrically as (x(t), y(t)) = (t, f (t)), meaning that the particle moves so that at time t it is above the point x = t with height f (t).
example: Suppose a particle starts at time t = 0 at the point (x(0), y(0)) = (1, 2) , and moves with constant velocity until time t = 1 to the point (x(1), y(1)) = (4, 6). The horizontal velocity is 41 −−^10 = 3, the vertical velocty is 61 −−^20 = 4,∗^ and the position at time t will be:
(x(t), y(t)) = (1 + 3t, 2 + 4t) for t ∈ [0, 1] ,
shown by the thick line segment below.
If we keep the same velocity for all real values of t, we get the thin infinite line.
Notes by Peter Magyar [email protected] ∗The overall speed is √ 32 + 4 (^2) = 5.
Given any parametric curve, writing it in terms of an equation satisfied by x and y is called deparametrizing: in this case, we want the graph of a linear function y = mx + b. A general method is to solve for t in terms of x, then plug in to the equation for y:
{ x = 1 + 3t y = 2 + 4t
t = 13 (x−1) y = 2 + 4
3 (x−1)
) =⇒ y = 43 x + 23.
Indeed, we could have immediately seen that the slope is the vertical velocity over the horizontal velocity: m = 43.
Parametric circles. Given the unit circle defined by the equation x^2 + y^2 = 1, we would like to parametrize it: to trace the curve by a particle moving according to (x(t), y(t)). One way is to let the particle make an angle of t radians at time t, meaning: (x(t), y(t)) = (cos(t), sin(t)) for t ∈ [0, 2 π].
If we keep the same motion for all t, the particle travels around and around the circle. We can check that this formula does trace the circle, because the coordinates do satisfy the known equation:
x(t)^2 + y(t)^2 = cos^2 (t) + sin^2 (t) = 1.
Our standard circular motion has center (0, 0), radius r = 1, starting at (1, 0) for t = 0, with 1 counterclockwise rotation during t ∈ [0, 2 π]. We can modify each part of this:
- Stretch the radius to r = 5: (x(t), y(t)) = (5 cos(t), 5 sin(t)).
- Move the center of the circle to (6, 7): (x(t), y(t)) = (6 + cos(t), 7 + sin(t)).
- Start from the bottom point (0, −1) at t = 0: (x(t), y(t)) = (cos(t− π 2 ), sin(t− π 2 )).
- Make the rotation clockwise:†^ (x(t), y(t)) = (cos(−t), sin(−t)).
- Do 10 rotations over t ∈ [0, 1]: (x(t), y(t)) = (cos(10· 2 πt), sin(10· 2 πt)).
†In general, to reverse the motion of (x(t), y(t)), take (x(−t), y(−t)) making time go backwards.
As the wheel rolls by t radians, its circumference traces an equal distance t along the x-axis, so the wheel’s center moves to (t, 1). If the center were fixed at the origin, the particle on the rim would start at (− 1 , 0) and turn clockwise once over t ∈ [0, 2 π], so its position would be:
(cos(−t− π 2 ), sin(−t− π 2 )) = (− sin(t), − cos(t)).
Adding the linear motion of the center and the circular motion around the center gives the parametric equation of the cycloid curve:
(x(t), y(t)) = (t − sin(t), 1 − cos(t)).
These equations allow us (or a computer) to easily plot the cycloid. Let us deparametrize this to get an xy-equation for the cycloid. We solve for t in terms of one variable (in this case y), and plug into the other variable (in this case x):‡
x = t − sin(t) y = 1 − cos(t)
cos(t) = 1 − y sin(t) =
1 −(1−y)^2 =
2 y−y^2 t = arccos(1−y) x = arccos(1−y) −
2 y−y^2
Simplifying:
cos
x +
y(2−y)
This weird xy-equation lets us easily check if a given point (x, y) lies on the cycloid. The parametric form, on the other hand, allows us to produce points on the curve.
Epicycloids. One variant of the cycloid is the epicycloid, in which the wheel rolls around a fixed circle. The curve varies depending on the relative size of the two circles. From the perspective of a fixed central Earth, the trajectories of the other planets are very close to epicycloids, and the classical astronomers in the tradition of Ptolemy attempted to find an exact model for planetary motion by adding further epicycles, wheels rolling on wheels like a gigantic clockwork. But starting with Copernicus, we interpret the apparent epicycloid as an illusion based on combining the separate orbits of Earth and the other planet around a fixed central Sun. Here is a compound epicycloid with a central circle of radius 1, a wheel of radius 1 6 rolling around it, and a wheel of radius^
1 6 rolling around that (assuming the circles can pass through each other):
‡Strictly, sin(t) = ±√ 1 − cos (^2) (t), but the minus sign just gives an extraneous upside-down cycloid.
