Triangle Area and Angles v1.3

Calculates the area and angles of any triangle described by 2D,3D..nD points

Description

This simple function use three points as input which can be 2D,3D...nD. And will calculate the area and angles of this triangle / polygon.

The last version also support the precise 'Orthogonal-triangular decomposition' area calculation method.

• MATLAB release: MATLAB 7.5 (R2007b)

[Scott Miller]

Angles will be returned in degrees.

Scott

[John D'Errico]

A couple of things to note here. First is as Scott noted, that triangle_angles has non-standard return for Matlab. It returns degrees, not radians. This is undocumented in the help, and should be repaired.

Triangle_area uses Heron's formula for the area of a triangle. Is this a good choice? The numerical analyst in me says to beware. Anytime that you square and square root some numbers, expect trouble. The problems will be seen when two of the triangle vertices are very near to each other. Try out this example:

format long g
pts = [0 0;0 1;1.e-9 0];

I've chosen this to be a right triangle, so the area is easy to compute. It will be exactly 5e-10.

triangle_area(pts)
ans =
      5.00000041370185e-10

As expected, we see a problem. Could this have been done more accurately? Its easy to do it for the 2-d case, but when you move to more dimensions, how might we solve the problem? This little code fragment will do it for you, in ANY number of dimensions, 2 or more. And it will have a high accuracy.

[q,r,e] = qr((pts(2:3,:) - repmat(pts(1,:),2,1))');
abs(prod(diag(r)))/2
ans =
                    5e-10

I used a pivoted QR here to ensure success even on nasty problems. Yes, its true that for most problems, there will be little difference. Its only on the nasty ones that you might worry.

pts = rand(3,10);
[q,r,e] = qr((pts(2:3,:) - repmat(pts(1,:),2,1))');
abs(prod(diag(r)))/2
ans =
         0.499463557359012

triangle_area(pts)
ans =
         0.499463557359011

But this is what numerical analysis does for you. It helps you to deal with those rare but nasty problems. Yes, triangle area was correct on the 10-d problem. How about this one, where the triangle is extremely flat, yet no two points are close to each other?

pts = [0 0;eps 1; 0 2];
triangle_area(pts)
ans =
     0

[q,r,e] = qr((pts(2:3,:) - repmat(pts(1,:),2,1))');
abs(prod(diag(r)))/2
ans =
      2.22044604925031e-16

In my choice of a rating for this submission, I considered these two flaws in the codes, as detailed above. Then I looked at the help. Its reasonable, except for the degrees versus radians problem. What the codes lack is error checks. Yes, there is only one input argument. It should be a numeric array, of size 3xn. Test for those properties. If the user has made a mistake in their use of the code, be nice and tell them that, rather than let them get possibly random results, or a confusing error message. I'd also recommend that the author explain what methods they have used. Yes, I recognized Heron's formula. But some other users might not do so.

The sum of these problems convinces me to rate it a 3. However when repairs are made, I'm always happy to change a rating.

[John D'Errico]

Now repaired, you can choose between methods, with the QR as the default. The only downside to the QR is a slight increase in time, by my check, .0004 seconds. So if you want to sacrifice some potential accuracy for a wee bit of time, you now have the option. The angles tool now also allows you to specify radians or degrees.

Its got error checks on the arguments, this submission looks ok to me now.

[Andres Toennesmann]

Thank you! Just as a suggestion:
You may add a new method that is a stable alternative to Heron's formula (see Wikipedia...) and that still works at about double speed compared to the orthogonal-triangular decomposition:

...
elseif strcmpi(method,'hs') % http://en.wikipedia.org/wiki/Heron%27s_formula, http://http.cs.berkeley.edu/~wkahan/Triangle.pdf
    L=[sqrt(sum((P(1,:)-P(2,:)).^2)) sqrt(sum((P(2,:)-P(3,:)).^2)) sqrt(sum((P(3,:)-P(1,:)).^2))];
    L=sort(L);
    % Area calculation with stabilized Heron's formula
    area = sqrt( (L(3)+(L(2)+L(1)))*(L(1)-(L(3)-L(2)))*(L(1)+(L(3)-L(2)))*(L(3)+(L(2)-L(1))) )/4;
...

Cheers
Andres

[Jane Guo]