Now that I review my own question more thoroughly, I'm not certain that is belongs in this area of the forums. My apologies. Perhaps it should be moved into University Trigonometry. I'm afraid I'm not sure.
Hello math enthusiasts and experts! I'm a software developer who has recently become enamored with animation on the web. Several of the posts on this forum have already proven extremely helpful in solving some of my earlier conundrums, but I think I've run into one that requires some specific assistance. As such, I've just now registered and this is my first post. Thank you in advance.
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I'm trying to do some programmatic animation, and I've been puzzling over the actual cartesian coordinates that will reside on the motion path. Here's what I want:
In a rectangular area with cartesian point of (0, 0) as its center point, my object (a circle) has a center point initially located at any coordinates that do not have a zero for either the X or Y value.
My object's final resting point (end of animation) will be a point mirrored to the quadrant that is vertically opposite of the one from whence the object originates. For example, if my starting point is (3, -2), then my end point will be (3, 2). If my starting point is (-5, 4), them my end point will be (-5, -4). As such, if my starting point is (x1, y1), my end point is (x1, -y1).
The midpoint of my object's path will be (0, 0).
The path that the object will traverse will be visually identical to that of one half of an ellipse.
Based on these requirements, I believe what I'm looking for is a way to calculate the Steiner circumellipse of the isosceles triangle defined by the start, mid, and endpoints of the animation.
I've been reading up on Steiner circumellipses, but I'm not certain I understand how to perform a formulaic calculation to determine such. To exacerbate the calculation, I wish to determine the cartesian position of my object based on the percentage of the animation that has completed.
Complete example: My animation start point is (40, 25). As specified above, my animation end point is therefore (40, -25) and my animation midpoint is (0, 0). At 0% of the way through the animation, my circle's center point coordinates will be those of the start point (40, 25). At 100% of the way through the animation, my circle's center point coordinates will be those of the end point (40, -25). At 50% of the way through the animation, my circle's center point coordinates will be (0, 0).
What I'm looking for is a way to solve for the correct cartesian coordinates of my circle's center point at p, where p is the percentage of the animation that has completed.
So if my start point is (40, 25), what are the correct coordinates for the circle when p=35? What about when p=96? what about when p=42?
Now that I review my own question more thoroughly, I'm not certain that is belongs in this area of the forums. My apologies. Perhaps it should be moved into University Trigonometry. I'm afraid I'm not sure.
My understanding of the Steiner circumellipse was flawed. The Steiner circumellipse is the ellipse of the smallest area that circumscribes the given triangle. What I want is the ellipse of the smallest area that circumscribes a parallelogram created my a mirroring of the aforementioned triangle.
Okay, so I'm getting closer, by which I mean I'm simplifying my requirements.
Given points (x, y) and (0, 0), I need to plot out the path of the ellipse that circumscribes a rhombus containing points (x, y), (0, 0), and (x, -y). Technically I only need to know the half of that path that circumscribes the three points. I should (and will) remove the mention of the Steiner Circumellipse from my original post's title.
Hi,
The attachment mostly solves your original problem. If x1<0, you need to change c and d appropriately. Also, you can add a parameter s in the parametric equations which controls the speed of the animation. If you have any question, post again and I'll try to answer. I successfully animated the figure in the attachment with my own software.
johng, thank you VERY much for your answer. After realizing that the Steiner circumellipse was not really what I was after, your illustration served as a very valuable jumping-off point for what would become my solution. I ended up using a parametric equation not unlike what you suggested. My work is not quite complete, but an early prototype of it is available for demonstration at this link: CodePen - Pen
Right now my circles are not technically colinear, as they are positioned based on the top left corner of a square that circumscribes them. Additionally, I have not yet properly limited my animation to only Pi radians of motion. This will be corrected later. Finally, the simple circles will be replaced with images of circles with no fill, and they will eventually make for a (hopefully) compelling moving lens flare effect on a web site I'm building.
Again, many thanks for your contribution.