Standard parametrizations

The simplest curve to parametrize is a straight line. A line through the origin along a vector ⟨a, b⟩ can be parametrized with

x = at,
y = bt.

If you want a ray (half-line) starting at the origin and travelling in the direction of this vector, then use the same formulas for x and y but add the restriction

t ≥ 0

for a closed ray (including the endpoint at the origin) or

t > 0

for an open ray (not including that endpoint). For a line segment running along the length of that vector, use the restriction

0 ≤ t ≤ 1

for a closed line segment (including both endpoints) or

0 < t < 1

for an open line segment (including neither endpoint). For both rays and line segments, the closed version is the usual standard, although there are times when the open version is needed instead.

If the line (or ray or line segment) doesn't go through the origin, then you'll need some point (x₁, y₁) that it does go through. Then you can use

x = x₁ + at,
y = y₁ + bt.

Again, without any restriction on t, this is a line; but you can restrict t as above to get a ray or a line segment. Or if you have two points on the line, then you can subtract them to get the relevant vector. Then the parametrization becomes

x = x₁ + (x₂ − x₁)t,
y = y₁ + (y₂ − y₁)t.

All of this works in any number of dimensions; the line through P₁ along the vector v has the parametrization

P = P₁ + tv,

and the line through P₁ and P₂ is

P = P₁ + t(P₂ − P₁).

The same restrictions on t as before will turn these into rays or line segments.

Going back to 2 dimensions, the unit circle (whose radius is 1 and whose centre is at the origin) is usually parametrized like this:

x = cos t,
y = sin t.

If there are no restrictions on t, then you are effectively going around and around the circle forever, counterclockwise (in a counterclockwise coordinate system). If you want the parametrization to be one-to-one, so that every point on the circle is covered exactly once, then you need a restriction on t; the usual one is

0 ≤ t < 2π.

It's even more common to use

0 ≤ t ≤ 2π;

this is almost one-to-one (since only the point (1, 0) is covered twice, once when t = 0 and once when t = 2π), and it has a compact domain (which is helpful for some things). So this restriction is the standard one for a circle.

If the radius of the circle is r, then the parametrization becomes

x = r cos t,
y = r sin t.

If the circle is centred at (h, k) instead of at the origin, then the parametrization becomes

x = h + r cos t,
y = k + r sin t.

You use the same restrictions as before to make the parametrization one-to-one or almost one-to-one.

Another useful parametrization is the graph of a function f. For this, you can use x itself as the parameter:

x = t,
y = f(t).

Since you can always call the parameter something else instead of t, you can even call it x:

x = x,
y = f(x).

If you only want the graph of the function restricted to an interval [a, b], then place this restriction on the parameter:

a ≤ t ≤ b

(or a ≤ x ≤ b if you are calling the parameter x instead of t). This works more generally any time you have an equation that you can solve for y; if you get a unique solution, then this equation defines a function, and the equation y = f(x) in the parametrization above is the equation that you get when you solve for y.

If you solve for x instead of for y, then you can say that x is some function g of y. This isn't the graph of that function exactly, since the variables come in the wrong order, but you can still parametrize the curve using y as the parameter:

x = g(t),
y = t.

Again, you can put a restriction on t if you only want certain values of the independent variable, which is now y.

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This web page was written by Toby Bartels, last edited on 2020 November 19. Toby reserves no legal rights to it.

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