coordinates of a point on the surface of a sphere that is closest to an external point.

[Aiswarya_M]

1. I want to find the coordinates of a point on the surface of a sphere that is closest to an external point (in the picture, the external point is START).
2. And I also want to find the set of coordinates/locus of the arc in the direction of the line and on the periphery of the sphere.
Kindly help. Thanks.

 

[Dr.Peterson]

The closest point on a sphere is the point directly between you and the center of the sphere. So you can just make the line from START to the center, and intersect it with the sphere. (It would be more obvious if you had an undistorted picture, as this is not true for an ellipsoid in general.)

I don't know what arcs you are asking about in the second part.

 

[Aiswarya_M]

My objective is to find the equations of the major and minor an arcs at the periphery of the sphere, passing through the intersection of the line (represented by the green dots) and the sphere.
And I want to find the coordinates of all the points in those arcs.
It is a sphere, not an ellipsoid (it appears distorted due to the lack of aspect ratio)

 

[Dr.Peterson]

An arc has two ends. I understand where one end is, but not the other! Is it the nearer intersection of the line through START, parallel to the y axis, and the sphere?

In what form do you want the answer?

What effort have you made to solve the problem?

 

[firemath]

So are you asking for the parabola function? Arc equation? I don't really understand what you are asking here.

If you need that "closest point" in a precise form, there is a simple distance formula calculation, but I doubt you need that.

 

[Dr.Peterson]

Let's look at the point first.

Suppose START is at P(x,y,z), and the center of the sphere is at C(a,b,c), with radius r.

Find the distance between C and P, and call that d. Form the vector v = CP, whose length is d; you want the vector u in the same direction with length r, which will be (r/d) v. Add that to the position vector for C, and you've found the point you want.

For the arc (an arc of a circle on the surface of the sphere), we need to know what you need to specify it. That depends on how you are planning to implement this. If you're going to draw it using some graphics package, for example, we need to know what input it will need. You said:

My objective is to find the equations of the major and minor an arcs at the periphery of the sphere, passing through the intersection of the line (represented by the green dots) and the sphere.
And I want to find the coordinates of all the points in those arcs.
It is a sphere, not an ellipsoid (it appears distorted due to the lack of aspect ratio)

We can't give you a list of all the points; we could give you an equation of a circle in various forms, together with an inequality to restrict that to one arc of the circle. Probably a parametric form would be most suitable. On the other hand, depending on your project, it is probably better to do what this site is meant to do, and help you learn enough to work out those details yourself, and understand what you are doing.

 

[Aiswarya_M]

I have found the equations of the sphere, the straight line and the points of intersection P and Q.
I want to find the equation of the circle in terms of x, y, z that:
* passes through the line
* is a part of the sphere
* lies in a plane normal to the X-axis

 

[Dr.Peterson]

The circle on the sphere that passes through P and Q is simply the intersection of the plane through C, P, and Q and the sphere.

This plane will not necessarily be normal to the x-axis; I'm not sure why you added that condition.

And you seem to have abandoned the idea of using the point on the sphere closest to P. Am I misunderstanding something, or have you changed the problem while introducing a "goal" that was not mentioned before?