Parametric circles and lines

Start from the unit circle: $(\cos t, \sin t)$ sits on it for every t, with t the angle from the positive x-direction. Scale by r and shift by (a, b):

1. $x = a + r\cos t$, $y = b + r\sin t$ traces the circle centered at $(a, b)$ with radius $r$.
2. The center comes from the added constants; the radius from the shared coefficient. In $x = 3 + 2\cos t$, $y = -1 + 2\sin t$: center (3, −1), radius 2.

Check by extremes: cos t and sin t live in [−1, 1], so x ranges over $[a - r,\, a + r]$ and y over $[b - r,\, b + r]$, exactly a circle's bounding box. And if the two coefficients differ, as in $x = 3\cos t$, $y = 2\sin t$, the bounding box is not square and the curve is an ellipse, not any circle.

The standard template starts at angle t = 0, the point $(a + r, b)$ due right of center, and sweeps counterclockwise: by $t = \pi/2$ it is at the top. But the template is a default, not a law. Swap the roles, $x = r\sin t$, $y = r\cos t$, and the walk starts at the top and runs clockwise. Negate the sine, $x = r\cos t$, $y = -r\sin t$, and it starts due right and runs clockwise.

So never announce "counterclockwise from the right" by reflex. Interrogate the formulas the same way as any parametric curve: evaluate at t = 0 for the start, then at a slightly larger t for the direction. Two evaluations settle everything.

The full circle needs a full $2\pi$ of parameter. Anything less traces an arc, and the arc is found by auditing the interval, never by trusting the eliminated equation. On $\pi/2 \le t \le \pi$, the unit-circle template runs from (0, 1) at the top to (−1, 0) at the left: a quarter circle in the second quadrant. The identity $x^2 + y^2 = 1$ still holds at every traced point, and still describes a whole circle the walk never finished.

Intervals longer than $2\pi$ re-cover ground: on $[0, 3\pi]$ the walk laps the circle once, then covers the top half a second time. The traced SET is the full circle; the journey visited some of it twice. Which answer matters depends on the question, so keep set and journey separate in your head.

To walk from $P_0 = (x_0, y_0)$ to $P_1 = (x_1, y_1)$ as t runs 0 to 1, aim each coordinate at its destination: $x = x_0 + t\,(x_1 - x_0)$, $y = y_0 + t\,(y_1 - y_0)$. At t = 0 the walk is at $P_0$; at t = 1, at $P_1$; in between it moves along the straight segment at constant pace. Always check both endpoints; the two most common build errors are using the destination coordinates as the step sizes, and letting a coordinate drift that should hold still.

As always, this is one description among many. Reversing the endpoints walks the same segment backward; replacing t by 2t walks it in half the parameter; and an unrestricted linear pair like $x = 1 + 2t$, $y = 3 - t$ for all real t is not a segment at all but an entire line. The set of points and the way they are walked are separate choices, and both are yours.