# Area of a surface spanned by 2 parametric curves

I would like to find the area of a surface spanned by two parametric curves in 3D. I came up with this metric for distance between curves.

The illustrations are in 2D for simplicity.

curveA[t_] := {0, 4} + t {3, -4}
curveB[t_] := {8, 2} + t {0, -2}
curveC[t_] := {2, 2} + t {0, -2}

Show[{ParametricPlot[{curveA[t], curveB[t]}, {t, 0, 1},
PlotStyle -> {Directive[{Red, Thickness[0.01]}]}],
Graphics[{EdgeForm[None], FaceForm[Lighter[Blue]],
Polygon[{{0, 4}, {3, 0}, {8, 0}, {8, 2}}]}]}, Axes -> True]
Show[{ParametricPlot[{curveA[t], curveC[t]}, {t, 0, 1},
PlotStyle -> {Directive[{Red, Thickness[0.01]}]}],
Graphics[{EdgeForm[None], FaceForm[Lighter[Blue]],
Polygon[{{0, 4}, {2, 2}, {2, 0}, {3, 0}}]}]}, Axes -> True] The areas are 18 and 4/3, repectively, if I am not mistaken.

A more general 3D non-intersecting case. The "orientation" of the curves are the same, but they can intersect each other. Thank you very much!

PS: If this issue is more appropriate in the math stack exchange hub, please move this thread.

• In the second 2D example the area is 5/3 since the intersection point doesn't correspond to the same parameter on the 2 lines - so the drawing is misleading as well. Thanks nikie! – Toorop Aug 31 '15 at 14:05

If we define the 2d cross product like this:

cross2d[a_, b_] := a.{{0, 1}, {-1, 0}}.b


I think it's just:

Integrate[(Abs[cross2d[curveA'[t], curveB[t] - curveA[t]]] +
Abs[cross2d[curveB'[t], curveB[t] - curveA[t]]])/2, {t, 0, 1}]


Where the idea is that cross[curveA[t]-curveA[t+d], curveB[t] - curveA[t]]/2 is the area of the triangle {curveA[t], curveA[t+d], curveB[t]}. Add the area of the triangle {curveA[t], curveB[t+d], curveB[t]}, and you get the area of the quad {curveA[t], curveA[t+d], curveB[t+d]}, curveB[t]}. Let d go to 0, and you get the integral above.

for a pair of 3d curves, it would be:

Integrate[(Norm[Cross[curveA'[t], curveB[t] - curveA[t]]] +
Norm[Cross[curveB'[t], curveB[t] - curveA[t]]])/2, {t, 0, 1}]