Intersection of line and surface

In summary, a straight line in 3 space can be represented as A + Bt, where A is the position, B is the direction, and t is a scalar parameter. CAD surfaces can be described using polynomial functions of two variables (u and v) with the highest degree term being u^nv^n. The intersections can be found by solving a polynomial in t. For n = 2 or n = 3, the polynomial in t is of 8th or 18th degree respectively (2n^2). It appears that this relationship holds for n > 3 as well, although there is no proof at this time. It has also been observed to work for n = 1.
  • #1
mathman
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A straight line in 3 space can be described as A + Bt, where A is a position, B a direction, and t a scalar parameter. CAD surfaces can be represented in terms of polynomial functions of two variables (u and v) with the highest degree term being [itex]u^nv^n[/itex]. The intersections can then be obtained as roots of a polynomial in t. I have seen proofs that for n = 2 or n = 3, the polynomial in t is of 8th or 18th degree respectively [itex](2n^2)[/itex].

Question: Does this relationship [itex](2n^2)[/itex] hold for n > 3?
 
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  • #2
I do not have a proof but it looks like the general formula.

It also works for n=1.
 

1. What is the definition of intersection of line and surface?

The intersection of a line and a surface is the point, set of points, or curve where the line and surface meet or cross each other.

2. How is the intersection of line and surface calculated?

The intersection is calculated by setting the equations of the line and surface equal to each other and solving for the variables that represent the point of intersection.

3. Can there be more than one intersection of a line and surface?

Yes, there can be multiple intersections between a line and a surface, depending on their equations and the shape of the surface. It is also possible for there to be no intersection at all.

4. What does the intersection of a line and surface represent in terms of geometry?

The intersection of a line and surface represents the points where the line lies on the surface or intersects with it. It can also represent the solution to a system of equations involving the line and surface.

5. How is the intersection of line and surface used in real-world applications?

The intersection of line and surface is commonly used in fields such as engineering, architecture, and computer graphics to determine the position and orientation of an object in 3D space. It is also used in mathematical modeling to solve equations and analyze geometric relationships.

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