Topic 4.4 Notes – Parametrically Defined Circles and Lines

What Parametric Circles and Lines Are

A parametric path has the form $(x(t),y(t))$. As $t$ changes, the point $(x,y)$ moves.

That means a parametrization gives you two things at once:

• the geometric path itself, like a circle or a segment
• the motion along the path, including
  • starting point
  • ending point
  • direction
  • how much of the path is covered
  • how the coordinates change as $t$ changes

This is why the equations and the domain have to be read together. The same formulas with a different interval can give one arc, a full circle, or multiple trips around it.

Also, many different parametrizations can represent the same shape. So on a quiz, don’t stop at “it’s a circle” or “it’s a line.” Check what happens at the beginning, middle, and end of the parameter interval.

Circles written with a parameter

The standard unit circle uses

$$(x(t),y(t))=(\cos t,\sin t)$$

for $0\le t\le 2\pi$.

That traces one full counterclockwise revolution starting at $(1,0)$, then going through:

• $t=0 \rightarrow (1,0)$
• $t=\frac{\pi}{2} \rightarrow (0,1)$
• $t=\pi \rightarrow (-1,0)$
• $t=\frac{3\pi}{2} \rightarrow (0,-1)$
• $t=2\pi \rightarrow (1,0)$

The unit circle diagram is a quick way to connect those parameter values to the usual angle positions and key coordinates.

A circle centered at $(h,k)$ with radius $r$ is

$$(x(t),y(t))=(h+r\cos t,\ k+r\sin t)$$

For $0\le t\le 2\pi$, it starts at the rightmost point $(h+r,k)$.

To change start, direction, or speed, use

$$(x(t),y(t))=(h+r\cos(\omega t+\phi),\ k+r\sin(\omega t+\phi))$$

• $\phi$ sets the starting angle
• $\omega>0$ means counterclockwise
• $\omega<0$ means clockwise
• one full revolution takes parameter length $\frac{2\pi}{|\omega|}$

A common clockwise form is $x(t)=h+r\cos t,\ y(t)=k-r\sin t$.

The domain tells you what part is traced:

• $0\le t\le \pi$ gives the upper semicircle
• $\frac{\pi}{2}\le t\le \frac{3\pi}{2}$ gives the left semicircle
• $0\le t\le 4\pi$ traces the circle twice

Building a Circle Model from a Description

When a word problem describes circular motion, pull out these features:

1. center
2. radius
3. starting point or starting angle
4. direction
5. time interval

If motion starts at $t=t_0$, angle form is clean:

$$\theta(t)=\alpha+\omega(t-t_0)$$

$$x(t)=h+r\cos(\alpha+\omega(t-t_0)),\qquad y(t)=k+r\sin(\alpha+\omega(t-t_0))$$

For one full revolution from $t_0$ to $t_1$,

• counterclockwise uses $\omega=\frac{2\pi}{t_1-t_0}$
• clockwise uses $\omega=-\frac{2\pi}{t_1-t_0}$

Then plug in the start time and one or two other values to make sure the point is where it should be and moving the right way. That check catches a lot of sign mistakes.

Parametrizing Line Segments

From $P_1=(x_1,y_1)$ to $P_2=(x_2,y_2)$, the standard form is

$$(x(t),y(t))=(x_1,y_1)+t(x_2-x_1,\ y_2-y_1), \quad 0\le t\le 1$$

Equivalent form:

$$\big((1-t)x_1+tx_2,\ (1-t)y_1+ty_2\big)$$

Here:

• $t=0$ gives $P_1$
• $t=1$ gives $P_2$
• $t=\frac12$ gives the midpoint

The graph below shows the same idea with a point moving along a straight path as $t$ changes.

If the motion goes from $t=a$ to $t=b$, use

$$x(t)=x_1+\frac{t-a}{b-a}(x_2-x_1), \qquad y(t)=y_1+\frac{t-a}{b-a}(y_2-y_1)$$

You can also write a line from an initial point and constant rates:

$$x(t)=x_0+u(t-t_0), \qquad y(t)=y_0+v(t-t_0)$$

Here $(u,v)$ are the horizontal and vertical rates.

The domain matters here too:

• $0\le t\le 1$ gives the segment
• $t\ge 0$ gives a ray
• all real $t$ gives the whole line

What to Check and Common Mistakes

For circles, check:

• center
• radius
• initial point
• direction
• total angle traveled

For lines, check:

• both endpoints
• one interior point such as the midpoint

Common mistakes:

• Using different angle expressions in sine and cosine breaks the circle.
• Center shifts go outside trig functions.
• Phase shifts go inside trig functions.
• Radius multiplies sine and cosine values.
• Different horizontal and vertical scale factors make an ellipse, not a circle.
• Forgetting the domain can turn a segment into a whole line or a single arc into repeated revolutions.