Parametric Circle equations and intersections

Question

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0. THE SHORT VERSION:

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How do you solve this kind of parametric system of equations ? Can we say it is linear ?

• Solving for $s,t$. We are given $r \in R$:

$$\cos t = \cos s \cdot r$$

$$\sin t = \sin s \cdot r$$

Solutions may exist in $Z$, maybe Euler Formula introducing $e^{i\theta}$ and $e^{i\phi}$ with $t=\theta$ and $s=\phi$ can be of some help here ?

Thanks a lot, longer version of the problem below if you wanna dive into it.

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Although having searched extensively on various boards about my issue, I couldn't seem to solve my problem which is the following:

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1. Defining the Goal

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Given two arbitrary ellipsis, I would like to identify the cases and find their 1, 2, 3, 4 or infinitely many intersection points. I would like to solve this using parametric equations of the ellipsis.

I can define my ellipsis with parametric equation in the 2D plane as such:

• Ellipsis A:
• A(t):
• $x(t) = \cos t \cdot a + h$
• $y(t) = \sin t \cdot b + k$
• $t$ is the parameter I want to find, $t \in [ 0, 2\pi ]$ or whatever
• $a$ is let's say the radius on the minor axis of the ellipsis A
• $b$ is let's say the radius on the major axis of the ellipsis A
• $( h, k )$ is the vector from origin to the center of my ellipsis A.

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• Ellipsis B:
• B(s):
• $x(s) = \cos s \cdot c + i$
• $y(s) = \sin s \cdot d + j$
• $s$ is the parameter I want to find, $s \in [ 0; 2\pi ]$ or whatever
• $c$ is let's say the radius on the minor axis of the ellipsis B
• $d$ is let's say the radius on the major axis of the ellipsis B
• $( i, j )$ is the vector from origin to the center of my ellipsis B.

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2. Apologies

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I'm not quite the mathematician myself, I'm doing this as a recreational exercise, and I wouldn't be too disappointed if it wasn't possible to solve but I think it is an interesting thing to think about.

One thing I didn't mention is the ellipsis could also be rotated by arbitrary angles theta & phi on top of that but solving without these rotations would be enough to satisfy my curiosity :)

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3. Trying to get everything done anyway

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Ok so now for the things I tried:

I know how to solve linear parametric systems for line intersections, but I understood that would not help me here because this system is not linear anymore.

One thing I tried is to simplify the cases up to trivial problems and try to solve adding known constants by iteration:

• Solving Intersection of two unit circles A & B at origin ( trivial )
• Solving Intersection of two unit circles A & B, A at origin and B with center at $( 1, 0 )$
• Solving Intersection of two unit circles A & B, A at origin and B with center at $( h, 0 )$, h known for the general case

These seemed linear

• Solving Intersection of two unit circles A & B, A at origin and B with center at $( h, k )$, $h$ & $k$ known for the general case

This one seemed linear too but I'm not sure of my solution

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4. The simplified case where part of the problem lies

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• Solving Intersection of two circles A & B, A unit circle at origin and B circle of Radius r centered at origin
• Solving Intersection of two circles A & B, A unit circle at origin and B circle of Radius r and B with center at ( h, k )

I could not solve these cases. I see these cases as special cases of my wider ellipsis problem, trying to solve one case at a time but I am already stuck at the point of the the two circles A & B at origin with A unit circle and B circle of Radius R. Solving this one will help me go further in this goal.

Defining parametric equations for this case gives us:

• Circle A:
• A(t):
• $x(t) = \cos t$
• $y(t) = \sin t$

• $t$ is the parameter I want to find, $t \in [ 0; 2\pi ]$ or whatever

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• Circle B:

• B(s):
• $x(s) = \cos s \cdot r$
• $y(s) = \sin s \cdot d$
• $s$ is the parameter I want to find, $s \in [ 0; 2\pi ]$ or whatever
• $r$ is the radius of the circle B, it is given.

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5. Formalisation

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• plugging $A = B$

• => $A( t ) = B( s )$

• $\cos t = \cos s \cdot r$

• $\sin t = \sin s \cdot r$

While it is geometrically trivial, it is analytically really hard to grasp, I don't have the tools to solve this, yet I am only doing the primary cases of my broader problem. The thing is I don't think i know the right tools in order to solve this. I guess my question, in the end, is the following:

• Is this solvable ?
• How to identify wheter an equation system is linear or not ?
• How to identify the polynomial degree of an equation system ? I guess ellipsis intersection dives into degree 4, yet circle equation are degree 2, how to know for sure ?
• What approach do you recommend in order to solve this
• Do you recommend working with parametric equations ? (I do have a weakness for them when working with lines, interpolations and geometry in general. They proved helpful in the case of Bézier curve manipulation & more.

Thanks for your time, I guess my question is pretty long but i wanted to be clear despite being straightforward. I don't have the kind of mind it takes to do advanced maths but i am really eager to understand how to do that kind of magic. Thanks again, looking forward to your responses.

Answer 1

First of all let rearrange your equations to $$x(t) = a\cdot cos t + h$$ and $$y(t) = b\cdot sin t + k$$ and for the second ellipse $$x(s) = c\cdot cos s + i$$ and $$y(s) = d\cdot sin s + j $$

In order to find the potential points of intersection you need to solve a system of four equations. $$a\cdot cos t + h =d\cdot cos s +j$$ , $$ b\cdot sin t + k= d\cdot cos s +j$$, $$ cos^2(t)+sin^2(t)=1$$ and $$ cos^2(s)+sin^2(s)=1$$ for s and t.

I strongly recommend numerical method for solving such a system.