Groin Vault: Development of Curves on Surfaces
 The lengths shown in the sketches are multiplied by the radius of the circle to determine the dimensions for an actual development.
 For any station x, equal to the length to be measured from the adjoining ridge line in plan, the corresponding dimension developed on the work is Arcsin x, producing the curve: y = Arcsin x

 DEVELOPMENT of ARCSIN CURVE: SAMPLE DATA
 x from Center Tangent to Circle Sheathing Angle Sheathing Angle Slope 1.00000 Undefined 90.00000° Undefined Undefined .99000 7.01792 81.97041° 7.08880 7.08881 .95000 3.04243 72.65901° 3.20256 3.20256 .90000 2.06474 66.44809° 2.29416 2.29416 .80000 1.33333 59.03620° 1.66667 1.66667 .70000 .98020 54.46780° 1.40028 1.40028 .60000 .75000 51.34019° 1.25000 1.25000 .50000 .57735 49.10660° 1.15470 1.15470 .40000 .43644 47.49432° 1.09109 1.09109 .30000 .31449 46.35044° 1.04829 1.04828 .20000 .20412 45.58467° 1.02062 1.02062 .10000 .10050 45.14395° 1.00504 1.00504 Zero Zero 45.00000° 1.00000 1.00000

Solving and Checking the Arc Lengths
 The arc length of the developed curve equals the arc length of the ellipse. Transposing the co-ordinates about the axes of the graph of the arcsin curve results in the graph of the sinusoid below.

The ellipse may be defined in terms of a Parametric Angle, φ.

Integrating from zero to π/2 with respect to φ returns an arc length of 1.91010.

Back to Main Page