The elliptical curve is a commonly studied and employed curve. Many things in nature, architecture, and science employ the use of elliptical curves.
They are typically studied in college level geometry classes as well as undergraduate math classes. For this reason, you will most likely run into this curve often in your studies and work.
Ellipses can be defined by their major and minor axes, rotation, and points on the ellipse where it hits the X-Y plane. These points are called confluences.
To find the farthest points on an elliptical curve that are away from a given point on the curve, we will first have to define what a point is on an ellipse. Then, we will explain how to find the farthest points that are away from a given point on the ellipse.
Calculate the distances from (1, 0) to each point on the ellipse
Now that you have the points on the ellipse, you need to calculate the distances from (1, 0) to each point on the ellipse.
There are two ways to do this. The first is to use the distance formula for general geometric objects: d(p 1 , p 2 ) = √(x2 1 − x2 2 ) + (y2 1 − y2 2 )
Where p 1 is point (1, 0), and p 2 is any other point on the ellipse. You would then have to calculate the x and y coordinates of each point p 2 and add them up. This would be a lengthy process!
The second way is much easier. Remember when we were finding the points on the ellipse in Part Two? We found all of those points by taking determinants of matrices.
Find the points on the ellipse that are farthest away from (1, 0)
Now that you know how to find the points on the ellipse that are closest to (1, 0), finding the points that are farthest away is the opposite process.
To do this, you will first have to find the two foci of the ellipse, L1 and L2. The foci are where the curve intersects with the line connecting the two end points of the axis. You can then use these foci and plug them into a formula to find all of the other points on the ellipse.
The formula is: A = (B^2-C^2)/(2bc). This equation takes in B and C of either end point of the axis and gives you A, which is one point on the ellipse that is not at either end point.
These points are located at ±4/5 on the x-axis and ±2/5 on the y-axis
Now that you know how to find the points on the ellipse that are farthest away from the origin (0, 0), you can find the points that are farthest away from one point on the ellipse.
These points are located at ±4/5 on the x-axis and ±2/5 on the y-axis. You can use this fact to find all of the points on the hyperbola, which is a curve similar to an ellipse, but with sharp corners at both ends.
To find these points, first locate point A on the graph above. Then, locate point B and draw a line through point A that is parallel to the x-axis. Locate where this line intersects with the (-)side of point B and record this number. Do this for both (-)sides of point B.
This process can also be used to find points far from any point on an elliptical orbit
Imagine that the elliptical orbit is a circle and that the point (1, 0) is at the center of the circle. Then, in this case, all points on the elliptical orbit are farthest from the point (1, 0) at the center of the circle.
You can also imagine this scenario: there are two people walking around a park with a path that is surrounded by grass. One person is walking clockwise around the path, and the other person is walking counter-clockwise.
If these two people were to meet in the middle of the path at some point, then they would be standing exactly opposite each other. In this case, if one of them was standing on their feet instead of walking, they would be farthest away from each other.
The same concept applies to points on an elliptical orbit- they are farthest away from any point on the elliptical orbit that is at (0, 0).
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