Unit 7 · Topic 4.4 · Optional
Parametrically Defined Circles and Lines
Build circular and linear paths from geometric information, then use the parameter domain to control starting position, direction, and how much of the path is traced.
Learning Goals
- Parametrize the unit circle and transformed circles.
- Control circular starting position, orientation, and revolution time.
- Verify a circular parametrization using a trigonometric identity.
- Construct a parametrization of the segment joining two points.
- Distinguish a complete line from a restricted line segment.
1. Start with the Unit Circle
The standard counterclockwise unit-circle motion is
At \(t=0\), the point starts at \((1,0)\). As \(t\) increases through one full \(2\pi\)-radian cycle, the point travels counterclockwise and returns to \((1,0)\).
2. Use the Five Key Inputs
| \(t\) | \(\cos t\) | \(\sin t\) | Position |
|---|---|---|---|
| \(0\) | \(1\) | \(0\) | \((1,0)\) |
| \(\pi/2\) | \(0\) | \(1\) | \((0,1)\) |
| \(\pi\) | \(-1\) | \(0\) | \((-1,0)\) |
| \(3\pi/2\) | \(0\) | \(-1\) | \((0,-1)\) |
| \(2\pi\) | \(1\) | \(0\) | \((1,0)\) |
3. Transform the Unit Circle
A circle centered at \((h,k)\) with radius \(r>0\) can be parametrized by
4. Interpret Every Circle Parameter
| Feature | Effect |
|---|---|
| \(h\) | Horizontal coordinate of the center |
| \(k\) | Vertical coordinate of the center |
| \(r\) | Distance from the center |
| \(t\) | Angle that orders positions around the circle |
| \([0,2\pi]\) | One complete counterclockwise revolution |
5. Circle Example: Center \((-2,3)\), Radius 4
The motion begins at \((2,3)\), then passes through \((-2,7)\), \((-6,3)\), and \((-2,-1)\) before returning to \((2,3)\). These are the right, top, left, and bottom points of the circle.
6. Verify the Circular Path
From the component equations,
Square and add:
The rectangular equation confirms the center and radius, while the parametric form preserves orientation.
7. Reverse the Circular Direction
Negating the angle reverses the orientation. Since cosine is even and sine is odd, a clockwise version with the same starting point is
The geometric circle is unchanged; only its traversal direction changes.
8. Change the Starting Point with Phase
The more flexible form
starts at the angle \(\phi\) when \(t=0\). The sign of \(\omega\) controls orientation, and \(|\omega|\) controls how quickly the angular input changes.
For example, \(\phi=\pi/2\) starts at the top point \((h,k+r)\).
9. Match One Revolution to a Time Interval
To complete one counterclockwise revolution in \(T>0\) seconds, use
Replacing \(2\pi/T\) with \(-2\pi/T\) reverses the motion. The parameter domain is part of the model: extending it beyond \(T\) begins another revolution.
10. Build a Segment Between Two Points
Let \(P=(x_1,y_1)\) and \(Q=(x_2,y_2)\). A direct parametrization from \(P\) to \(Q\) is
Equivalently,
The weights \(1-u\) and \(u\) add to 1, so the point interpolates between the two endpoints.
11. Segment Example: \(P=(-3,2)\) to \(Q=(5,-2)\)
12. Check Fractional Positions
| \(u\) | Position | Meaning |
|---|---|---|
| \(0\) | \((-3,2)\) | Start at \(P\) |
| \(1/4\) | \((-1,1)\) | One quarter of the way |
| \(1/2\) | \((1,0)\) | Midpoint |
| \(3/4\) | \((3,-1)\) | Three quarters of the way |
| \(1\) | \((5,-2)\) | End at \(Q\) |
13. Use Actual Time Instead of a Normalized Parameter
If the trip from \(P=(-3,2)\) to \(Q=(5,-2)\) takes 4 seconds at constant component rates, replace \(u\) by \(t/4\):
The horizontal position changes by 2 units per second and the vertical position by \(-1\) unit per second. At \(t=4\), the formulas produce \(Q\).
14. Segment, Ray, or Complete Line
| Parameter domain | Geometric result |
|---|---|
| \(0\le u\le1\) | Segment from \(P\) to \(Q\) |
| \(u\ge0\) | Ray beginning at \(P\) and passing through \(Q\) |
| All real \(u\) | Complete line through \(P\) and \(Q\) |
| \(1\ge u\ge0\) traversed in decreasing order | Same segment from \(Q\) to \(P\) |
Changing the domain can change the geometric object even when the component formulas remain the same.
15. AP Workflow and Common Errors
- Identify whether the path is circular or linear.
- For a circle, extract the center, radius, start point, orientation, and time interval.
- Begin with cosine for horizontal displacement and sine for vertical displacement.
- Use phase and angular rate only after the basic circle is correct.
- For a segment, calculate \(Q-P\) component by component.
- Substitute both domain endpoints to verify the required start and end.
- Check an intermediate value and, when useful, eliminate the parameter to confirm the path.
- Do not omit the center shifts \(h\) and \(k\).
- Do not use \([0,2\pi]\) when \(t\) represents a different revolution time without rescaling the angle.
- Do not reverse \(Q-P\) unless the intended starting point also changes.
- Do not omit the parameter domain; it distinguishes a segment from a full line.
- Do not assume identical curves have identical orientation.
Key Takeaways
- Construct parameterizations of lines and circles and interpret their orientation.
- Core relationship: \(x=h+r\cos t,\quad y=k+r\sin t\)
- Error check: Do not omit the center shifts h and k.
- Parametrize one counterclockwise revolution of the circle centered at \((2,-1)\) with radius 3.
- Modify that parametrization so it starts at the top and moves clockwise.
- Construct a circle model that completes one revolution in 8 seconds.
- Parametrize the segment from \((-2,5)\) to \((4,-1)\) using \(0\le u\le1\).
- Rewrite the segment model for a trip lasting 3 seconds and verify its midpoint position.