Talk:Line-plane intersection

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Stub Class Low Priority  Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Please also add comments to suggest improvements to the article.
To-do list for Line-plane intersection: edit · history · watch · refresh
  • The calculations assume that the line and plane intersect at a point. This is not always the case (they might not intersect, or they might intersect identically; the line might be a subset of the plane). These cases should also be covered.
  • Add additional material from this version: [1].

My recollection of matrix mathematics is a little hazy, but the bit I added:

\begin{bmatrix} x_a - x_0 \\ y_a - y_0 \\ z_a - z_0 \end{bmatrix} = \begin{bmatrix} x_a - x_b & x_1 - x_0 & x_2 - x_0 \\ y_a - y_b & y_1 - y_0 & y_2 - y_0 \\ z_a - z_b & z_1 - z_0 & z_2 - z_0 \end{bmatrix} \begin{bmatrix} t \\ u \\ v \end{bmatrix}

...should be correct. ~~ jim d

Also, I'd be more comfortable seeing the vectors referred to in AB, BC -style notation, eg. : J − A = tJK + uAB + vAC or perhaps something like P_J - P_A = t \times V_{JK} + u \times V_{AB} + v \times V_{AC} which is more sane in the algebraic sense than the former. Can someone who routinely uses these kind of formulae please make a call? ~~ jim d

[edit] no sense whatsoever

I have not taken calculus or any higher maths, but the variables don't seem to be labeled. Most people, specifically me, have no idea what p is, so to say that p = X1 and Y1 means nothing. LFStokols 01:23, 9 March 2007 (UTC)LFStokols

[edit] Amateur article

In its current form, this article seems naive and amateurish. It's weak mathematically, and not very convincing practically. As an example of the first, it overlooks the method in the article on Plücker coordinates:

Given a plane with equation
0 = a^0x_0 + a^1x_1 + a^2x_2 + a^3x_3 , \,\!
or more concisely 0 = a0x0+a⋅x; and given a line not in it with Plücker coordinates (d:m), then their point of intersection is
(x0 : x) = (a⋅d : a×m − a0d) .
The point coordinates, (x0:x1:x2:x3), can also be expressed in terms of Plücker coordinates as
x_i = \sum_{j \ne i} a^j p_{ij} , \qquad i = 0 \ldots 3 . \,\!

As an example of the latter, solving the 3×3 linear system with an explicit inverse is a dubious choice.

How are the objects presented? (Two points, three points, implicit equations, parametric equations, Plücker coordinates, …) Do we really want a line and plane? How about a line segment? A ray? A polygon in a plane? Are we supposed to determine the point of intersection? What if the line is parallel to the plane (or nearly so)? What if the line is in the plane? Are we doing pure mathematics, or numeric computations? If the latter, we have a trade-off between speed and reliability.

The more common representation for a plane is as an implicit equation, often standardized as a unit normal, u, and a distance from the origin, d.

\bold{u}\cdot\bold{p} - d = 0 \,\!

If we have a parametric equation for the line,

\bold{p}(t) = \bold{p}_0 + t (\bold{p}_1 - \bold{p}_0) , \,\!

then we may substitute to obtain a linear equation in t, easily solved without matrices.

I'm not prepared to touch this today, but maybe someone can use these pointers. --KSmrqT 06:30, 7 April 2007 (UTC)

I've merged in the other article. I think that its probable better named as Line–plane intersection than Line-plane intersection, but I don't have permissions to move it the other way. --Salix alba (talk) 09:22, 7 April 2007 (UTC)