Find a parametrization for the circle (x-2)^2+y^2=1 starting at (1,0) and moving clockwise once

Find a parametrization for the circle (x-2)^2+y^2=1 starting...

Key Concepts

Central Angle Parameterization

Using the central angle as the parameter means that the angle measured from the center of the circle to a point on the circle is used to determine the position of that point. This choice directly ties the geometrical interpretation of the angle to the standard trigonometric functions used in the parametrization.

Trigonometric Functions

Sine and cosine functions are fundamental in parametrizing circles due to their periodic properties and the Pythagorean identity. They allow for a smooth and continuous description of circular motion by capturing both the horizontal and vertical components.

Direction of Traversal

This concept involves determining whether the parameterization traces the circle in a clockwise or counterclockwise direction. Adjusting the sign in front of the sine function (or cosine function) can reverse the direction of the parametrization to match the desired movement.

Circle Parametrization

This concept refers to representing a circle using one or more equations that depend on a parameter, typically an angle. The usual method is to use trigonometric functions such as cosine and sine to express the x and y coordinates, which traces out the circle as the parameter varies.

Translation of a Circle

When a circle is not centered at the origin, its parametrization must be adjusted by translating the coordinates. This involves adding constants to the standard cosine and sine functions to account for the shift in the center of the circle.

Transcript

00:01 So in this question, we're asked to parameterize x minus 2 squared plus y squared equals 1.

00:10 And there's some more details, but we'll mention this later.

00:14 Let's first draw this guy.

00:16 So this is a translated circle.

00:18 In fact, it's moved two places or two units in the positive x direction.

00:25 So the circle we get is centered at 2 .0 and has radius 1 .5.

00:31 So it looks something like this.

00:36 This is an awful circle.

00:38 I hope you will bear with me for that.

00:43 So the other parts of the question are we have to start over here and parameterize it clockwise.

00:54 So the easiest way to parameterize circles is using the interior angle.

01:00 So let's draw that in.

01:04 So we're going to use this guy as a parameter.

01:09 Just a quick note, how can you tell it's a circle centered at 2? so x minus 2, that means that if we plug in 2, it behaves like 0.

01:22 So x squared plus y squared equals 1 is the greatest 1 circle centered at 0.

01:27 So if it says x minus 2, to get like the same value for x minus 2, it used to be x, you need to plug in a x value that's too higher.

01:40 So the center isn't at zero zero, it's two x is higher.

01:44 It's at two zero.

01:46 So that's how you figure out the translation.

01:49 So let's figure out the parameterization.

01:56 So normally when you get a circle, you'd say something like x equals cosine of t and y equals sine of t.

02:09 The problem is, so let's encode this in red, that it would start over here.

02:14 Here, well, not quite yet.

02:17 We need to move it around and goes that way...

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