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I have a seemingly simple linear algebra problem which I have trouble with and I would like to ask for some advice how to solve it. Here is the problem:

Given is a circle on the unit sphere:

[itex]

x^2 + \mu_s^2 + z^2 = 1

[/itex]

where [itex]\mu_s[/itex] is a known constant. So the circle lies in the x z plane at a given height.

Also given is an arbitrary vector [itex]v[/itex].

The problem is to find the vector function [itex]s(\nu)[/itex] with [itex]\nu \in [-1,1][/itex] such that the dot product between the solution and vector [itex]v[/itex] equals [itex]\nu[/itex]:

[itex]

xv_x + yv_y + zv_z = \nu

[/itex]

Here are my thoughts about this:

It is clear that a solution is not always defined for the whole range of [itex]\nu[/itex] and that for a given [itex]\nu[/itex] there can be 2 solutions. So I expect a quadratic formula to pop up. Also since the solution is on the circle I expect a sinus of some angle (or [itex]1-cos^2[/itex]) to be there as well.

The way I tried to approach it was to find an expression for [itex]x[/itex] and [itex]z[/itex]:

[itex]x=\sqrt{1-\mu_s^2-z^2}[/itex]

[itex]z = \frac{\nu-xv_x-\mu_s y}{v_z}[/itex]

and substitute z within the expression of [itex]x[/itex] which after some massaging gives me:

[itex]x^2 - \frac{2\nu \mu_s v_y v_x}{v_z^2 - v_x^2}x - \frac{(1-\mu_s^2)v_z^2 + \nu + \mu_s v_y}{v_z^2 - v_x^2}=0[/itex]

I get [itex]x_1[/itex] and [itex]x_2[/itex] from solving the quadratic formula and insert it into the formula I got for [itex]z[/itex]. The problem here is that I need to divide by [itex]v_z[/itex] which means there is no solution if [itex]v[/itex] lies in the [itex]xy[/itex] plane. I don't understand why this is geometrically.

However, the problem I now have is that if I check my solutions they don't lie on the given circle and the angle doesnt satisify the given constrain. So I must be doing something wrong and my hope is that somebody here can point me into the right direction. Does the overall approach make sense? Is there something I miss? Is there a different way to solve the problem?

Any comment is much appreciated.

Thanks,

David

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# Simple linear algebra problem (points on circle for a given vector and angle)

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