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Our First Question

7

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I.	Given k lines and 4-k general spheres in R3, how many common tangents to the spheres also meet the fixed lines?

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Such common tangents and transversals satisfy
One quadratic Plücker equation,
k linear equations to meet the lines, and
4-k quadratic equations of tangency.

We expect 2 24-k common transversals and tangents.

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Number of Transversal Tangents

k lines 4 3 2 1 0 4-k spheres 0 1 2 3 4 Expected Number 2 4 8 16 32