Talk:Line–plane intersection

Matrix equation added
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. : $$J - A = t JK + u AB + v AC$$ 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 — Preceding unsigned comment added by 217.43.171.129 (talk • contribs) 17:26, 3 October 2006

Why even bother?
Why bother even writing the section for "Algebraic Form" if you are going to use a ton of undefined variables? You can't just make up random variables and expect anyone to follow it. Either define them and do it right, or don't edit the article. —Preceding unsigned comment added by 24.18.158.132 (talk) 08:20, 5 August 2009 (UTC)

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

-- I agree... there are things like xa and xb with no explanation for what 'a' and 'b' are! —Preceding unsigned comment added by 66.169.236.186 (talk) 03:35, 18 January 2010 (UTC)


 * I read the above comments, and decided to have a look at the article to see what the problem was. I thought that, if variables were undefined as suggested, I would see if I could help by adding definitions. I have looked both at the current version, to which 66.169.236.186 refers, and at the version which was current on 9 March 2007, to which LFStokols was referring. In both cases at the start of the section "Parametric form" the relevant notation is explained quite unambiguously. Thus we are told that $$\mathbf(x_a, y_a, z_a)$$ and $$\mathbf(x_b, y_b, z_b)$$ are two distinct points along the line. I can understand that to someone unacquainted with mathematical notation it might not be clear that this means there are two points of which the coordinates are respectively $$\mathbf(x_a, y_a, z_a)$$ and $$\mathbf(x_b, y_b, z_b)$$. However, unfortunately it is not realistic to expect every article on a mathematical topic to explain every bit of the relevant mathematical background, terminology, notational conventions, etc. If we did so then many articles would expand to the size of text books, and they would become less comprehensible, because the essential points would become obscured in all the lengthy explanation of side issues. While I fully sympathise with anyone who finds it frustrating to be unable to understand an article, it is an unfortunate fact that in mathematics there are complex inter-relations among topics, and it is not possible to write a concise article which successfully does the job of several years' study of mathematics. It is very unfortunate but true that if anyone does not understand that saying that "$$\mathbf(x_a, y_a, z_a)$$ is a point on the line" is defining what xa, ya and yb mean, then they probably lack the necessary background to follow the article even if this information were spellt out. JamesBWatson (talk) 22:29, 24 January 2010 (UTC)
 * Even simple math on Wikipedia uses notation from calculus and function notation. We cover issues of chemistry while maintaining the ability to be understood by most non-chemists, it might be advantageous to spell it out so that more people would have free access to this level of information without the necessity of . I am a game developer, and this very problem has always been my sorest spot, math-wise, yet I am always doing math. I think we can definitely implement more explicit explanations to the subject of math. I'm not sure we need artificial constraints on who should be able to understand the information written here. At the very least, a subscript link to a page explaining Wikipedia's usage of mathematical notation would be helpful. 02:29, 23 September 2014 (UTC) — Preceding unsigned comment added by 98.148.128.154 (talk)
 * While there is a case for many mathematical articles to be made more accessible, I'm not convinced there is much which can be done here. Looking at a few other articles on the topic they all have a similar treatment involving vector notation and calculations with vectors. I do think the algebraic form is a little easier to follow and I've switched the order and tried to make the section a little clearer. But as Euclid is said to have explained to Ptolemy there is no Royal Road to geometry. --Salix alba (talk): 05:37, 23 September 2014 (UTC)

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.
 * $$ \mathbf{u}\cdot\mathbf{p} - d = 0 \,\!$$

If we have a parametric equation for the line,
 * $$ \mathbf{p}(t) = \mathbf{p}_0 + t (\mathbf{p}_1 - \mathbf{p}_0), \,\!$$

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)

Algebraic Form Inconsistent with Wikipedia Plane Definition
I didn't find the algebraic form too hard to follow, but there is one very sad thing: the definition of d ends up inverted relative to the one given on the http://en.wikipedia.org/wiki/Plane_(geometry) page. This is very easy to miss and is an unpleasant bug to track down in code. I think this is why it is important to use normal forms when possible.

I suggest that the first equation in the Algebraic Form section should be changed to 'p dot n + d == 0', and a minus sign added to d in subsequent equations. Or else an explicit mention should be made somewhere that the plane definition (in terms of coefficients a, b, c, and d) being used is different than the one on the Plane page. —Preceding unsigned comment added by 137.229.17.63 (talk) 21:09, 8 September 2009 (UTC)

I also notice that in Oct. 2008 there was a revision that removed a minus sign from the last equation for t. So it appears that this article has confused intslef over this issue in the past as well.

Bkerin2 (talk) 18:55, 9 September 2009 (UTC)

I went ahead and changed this, hope its all correct. Bkerin2 (talk) 19:20, 9 September 2009 (UTC)

I'd prefer it the original way. I've used this page a few times for my own software and it's confusing and prone to introduing bugs to have the double negatives i.e. "d = -(something)" then later everything uses "-d" which effectively removes the negative, showing that the typical definition of a plane in terms of "ax + by + cz + d = 0" is less natural than "ax + by + cz = d" i.e. "ax + by + cz - d = 0". The currently written equation for t, for a vector v, where l_a is the origin,


 * $$t={-d \over \mathbf{v}\cdot\mathbf{n}}$$

expands to


 * $$t={-(-\mathbf{p}_0\cdot\mathbf{n}) \over \mathbf{v}\cdot\mathbf{n}}$$

And that's where I make my mistakes, because of the double negative. The geometry is simple to do from scratch, but to read the article and use it is difficult, when I want it to be the opposite - I should be able to read this article and get the answer correct (and bug free!) without having to do it from scratch every time! My two cents worth... —Preceding unsigned comment added by 130.102.158.15 (talk) 23:39, 16 June 2010 (UTC)

This would correspond to the definition of a plane as stated in the section above,


 * $$ \mathbf{u}\cdot\mathbf{p} - d = 0 \,\!$$

as opposed to the current article definition of


 * $$\mathbf{p}\cdot\mathbf{n}+d=0$$. —Preceding unsigned comment added by 130.102.158.15 (talk) 23:47, 16 June 2010 (UTC)

Pseudocode
Some pseudocode, that would be nice. For the programmers (not mathematicians) among us. -- 41.204.111.27 (talk) 14:05, 28 November 2008 (UTC)

Changed Algebraic Form
Since I was confused by the old section at first and saw that there had been comments by others here as well, I rewrote the algebraic form section in a way that I feel is much clearer than the old one. For one thing, the old section had the scalar $$d$$ in the vector equation for a plane which didn't make much sense to me. With so much confusion over its sign, why not just get rid of it altogether? I also defined all the variables I used. There's also no reason to expand the answer out at the end; it just makes the article look messy and harder to read. This should especially be clearer for people who are looking to use this equation in their code (which I bet is a large % of people reading the article) as its just a straight substitution of the vectors in their code. Hopefully others agree that the new section is clearer. If you have any major problems with it, please reply to this. MochaFlux (talk) 09:13, 10 September 2010 (UTC)

Simpler Solution
So why not express the plane as the actual equation with ABCD values, not by the point in the plane, then the simpler solution for d will be:

d = - (A*P_x + B*P_y + C*P_z + D) / (A*v_x + B*v_y + C*v_z)