Parameterization of an ellipse?

[ImaLooser]

Suppose I have a vector space. There is circle with center at the origin of the vector space. There is also a line L going through the origin at some angle. On the circle is a point moving around the circle at a constant speed. The vector from the center to the point makes some angle with the line L. What is that angle as a function of time? I'm too dumb to figure it out.

 

[HallsofIvy]

A vector space does not have an "origin", it has a zero vector. You seem to be confusing "vector space" with coordinate system. Also you have titled this "parameterization of an ellipse" yet there is no ellipse in your post. That makes this difficult to understand!

If I do understand it correctly, you write the coordinates of the point moving around the circle $(x, y)= (r cos(\theta), r sin(\theta))$. One can show that "moving around the circle at constant speed" (which I interpret as meaning the same distance on the circumference in the same time) is the same as moving at a constant angular velocity (the same angle in the same time). That is, $\theta= \omega t$ where $\omega$ is the constant angular velocity.

That means we have $(x, y)= (r cos(\omega t), r sin(\omega t))$. If the line, L, makes angle $\phi$ with the positive x-axis (the slope of the line is $tan(\phi)$) then (x, y) makes angle $\theta- \phi= \omega t- \phi$ with line L.

 

[ImaLooser]

A vector space does not have an "origin", it has a zero vector. You seem to be confusing "vector space" with coordinate system. Also you have titled this "parameterization of an ellipse" yet there is no ellipse in your post. That makes this difficult to understand!

Aha. Suppose you are aligned with line L. That is, the origin of the coordinate system is in the center of your belly and line L is going out of the top of your head. The circle will look to you like an ellipse. So I thought that might be reflected in the solution somehow.

 

[meldraft]

I'm not sure I follow your reasoning, but the only way to make a circle appear like an ellipse in 2D is to stretch one of the axes. In 3D, the ellipse can be a projection of a rotated circle on the 2D plane.

 

[blue_raver22]

I understand that parameterization is a mathematical technique used to represent a geometric shape or object in terms of one or more variables. In this case, the ellipse is being parameterized using the angle between the vector from the center to the moving point and the line L as the variable.

To find the angle as a function of time, we can use trigonometric relationships and the properties of circles and ellipses. The angle between the vector and the line can be represented as the sum of two angles: the angle between the vector and the x-axis, and the angle between the x-axis and the line L.

Since the point is moving at a constant speed and the circle has a fixed radius, we can use the concept of angular velocity to determine the angle between the vector and the x-axis as a function of time. This can be represented as ωt, where ω is the angular velocity and t is time.

Next, we can use the properties of circles and ellipses to determine the angle between the x-axis and the line L. This angle can be represented as the inverse cosine of the ratio of the radius of the circle to the distance between the center and the line L.

Combining these two angles, we can express the angle between the vector and the line L as a function of time: θ(t) = ωt + cos^-1(r/d), where r is the radius of the circle and d is the distance between the center and the line L.

I hope this explanation helps in understanding the parameterization of an ellipse in this scenario. It is important to remember that mathematics can be challenging, but with perseverance and the use of relevant formulas and concepts, we can solve complex problems and gain a deeper understanding of the world around us.