Electric-Eye
conic section ellipse2
I expect most people have heard of the conic sections, the circle, elipse, parabola and hyperbola which can all be formed by slicing a cone with a plane. I've just learned of this geometric method which can be used to find the two foci which are a separate way of defining or constructing an ellipse.
It turns out that if you take two spheres, such that they are tangent to the surface of the cone on a circle, and also tangent to the plane slicing through the cube at one point, those points where each sphere is tangent to the plane will coincide with the foci necessary to construct the same ellipse by finding the curve that is the sum of the distance from any point on the curve to the two foci.
In this image, the outline of the cone and plane have been removed, leaving only the spheres, and the area of the ellipse with its curve and foci/tangent points of the spheres marked with small green spheres.
I think this picture helps you to appreciate that as the two spheres get closer to one another, the points at which they are tangent to the ellipse come closer together, until at the point the two spheres are tangent to one another the foci of the ellipse will have converged into one, thus rendering the planar section of the cube a circle, with a single constant radius around this center point.
You know, I think I would really like to study geometry more deeply, best yet as the mathematics of computer graphics. But I would want to study it à la carte, just choose whichever topics seem interesting and fruitful to me, without having to do things in the order someone else decided was best. No prerequisites either. Because if I have to know the math before I can use it then I will never use most of it, because I won't find most of it to be worth my time until I have an application to which it is an essential tool.
That's how I think math and physics should be taught, as the tools you need in order to construct your own video game.
conic section ellipse2
I expect most people have heard of the conic sections, the circle, elipse, parabola and hyperbola which can all be formed by slicing a cone with a plane. I've just learned of this geometric method which can be used to find the two foci which are a separate way of defining or constructing an ellipse.
It turns out that if you take two spheres, such that they are tangent to the surface of the cone on a circle, and also tangent to the plane slicing through the cube at one point, those points where each sphere is tangent to the plane will coincide with the foci necessary to construct the same ellipse by finding the curve that is the sum of the distance from any point on the curve to the two foci.
In this image, the outline of the cone and plane have been removed, leaving only the spheres, and the area of the ellipse with its curve and foci/tangent points of the spheres marked with small green spheres.
I think this picture helps you to appreciate that as the two spheres get closer to one another, the points at which they are tangent to the ellipse come closer together, until at the point the two spheres are tangent to one another the foci of the ellipse will have converged into one, thus rendering the planar section of the cube a circle, with a single constant radius around this center point.
You know, I think I would really like to study geometry more deeply, best yet as the mathematics of computer graphics. But I would want to study it à la carte, just choose whichever topics seem interesting and fruitful to me, without having to do things in the order someone else decided was best. No prerequisites either. Because if I have to know the math before I can use it then I will never use most of it, because I won't find most of it to be worth my time until I have an application to which it is an essential tool.
That's how I think math and physics should be taught, as the tools you need in order to construct your own video game.