Development of Curved Surfaces

Note: I have not been able to find any handy resources regarding development; the books on Amazon.com seem to be either out of print or expensive as hell, and the Web has, for once, failed to provide many answers. The methods I demonstrate here are my own, and may or may not be correct. However, I have used them, and they do indeed seem to work.

Development is the graphical process of producing a flat pattern from drawings of a curved surface. Not all surfaces lend themselves to development; any surface with a compound curve cannot be developed. I have used development in designing a boat and in sheet metal work, but I'm sure it is useful in other fields as well.

There are three methods of development; parallel line development, radial line development, and triangulation.

I. Parallel Line Development

Parallel Line Development uses geometry to measure a series of heights above a base line, and marks these heights off on a series of parallel lines. Often applied to pipe, it can be applied to any object with a constant cross section, as long as a view perpendicular to the cross section is available.

Basically it is nothing more than a series of lengths along the surface of a cylinder.

This is the plan we'll start with. No, I don't know what it is. We'll develop the surface of the small branch. Parallel Line Development plan.
1. Divide the plan view of the branch of the tee with a convenient number of equally spaced radii. I've used 9 radii because it produces eighths. Because this part is symetrical, I've only marked off half of it - I could actually only mark a quarter, since it's symetrical about two axes. Parallel Line Development step 1.
2. Extend perpendicular lines from the ends of each radii to the curved edge of the profile view of the branch. Parallel Line Development step 2.
3. From the points where the lines in step 2 meet the curved edge of the profile view, drop vertical lines down. Add an extra line where the lines cross the base line. Parallel Line Development step 3.
4. Measure the distance between the points where the radii meed the edge of the plan view, and mark off this distance on the vertical lines drawn in step 3. Draw horizontal lines through the marks. Parallel Line Development step 4.
5. Connect the dots where the corresponding lines cross. Parallel Line Development step 5.

II. Radial Line Development

Radial Line Development basically just Parallel Line Development for tapered objects - ie. cones, pyramids, etc. Alternately, Parallel Line Development is just Radial Line Development for an infinitely small taper.

See also my notes on Conic Frusta.

This is the plan we'll start with; a truncated cone with angle cut out. This is actually a design under consideration for the turret of my tank. Radial Line Development plan.
1. Divide the plan view into equally spaced radii. The more the better. As this object is symetrical, we only need to do this to half the plan - the other half will be a mirror image. Radial Line Development step 1.
2. Extend vertical lines from the points where these radii meet the circumference of the plan view upward to the base line of the profile... Radial Line Development step 2.
3. ...and then to the apex of the cone. This cone has a very shallow taper, so the apex is way up there. Radial Line Development step 3.
4. This particular design has one more point that has to be marked - the upper edge of the angled surface. Run a line from the apex through the edge of the angled surface to the base line of the profile, then vertical down to the circumference of the plan. It isn't necessary to continue the line to the center of the plan, but it looks nice. Radial Line Development step 4.
5. Draw horixontal lines from the points along the angled surface to the outermost radii on the profile view... Radial Line Development step 5.
6. ... and continue them as arcs around the apex. Radial Line Development step 6.
7. Measure the distance between the points where the radii meet the circumference in the plan view, and mark these distances off on the arc drawn in step 6 that cooresponds to the base line in the profile view. Radial Line Development step 7.
8. Connect the dots where the radii cross their cooresponding arcs. Remember that you've only got half a pattern here; the other side is a mirror image of this one. Radial Line Development step 8.

III. Triangulation

Triangulation is used for irregular surfaces. A good example is a boat hull. However, not all hulls can be developed - only those that do not have compound curves.

A little simple geometry is first required. The length of a line can be derived from the (X,Y,Z) coordinates of it's endpoints, using the following formula:

L = ((X1 - X2)2 + (Y1 - Y2)2 + (Z1 - Z2)2)½

If the distance LXY in the XY plane between the endpoints is known, this formula can be simplified to:

L = (Lxy2 + (Z1 - Z2)2)½

Assuming one point to be the origin simplifies this even further, and allows the Z dimension to be measured directly:

L = (Lxy2 + Z2)½

This last formula is used to calculate the distance between two points on a surface from the distance between them on the plan drawing and the height difference on the profile drawing.

Triangulation can be VERY tedious.

This is the plan we'll start with; a simple boat. Triangulation Development plan.
1. The first step is to draw vertical lines through the plan. I drew nine lines; the more lines, the more accurate the developed surface. It is not necessary for the lines to coincide with the extreme front and back of the plan. Triangulation Development step 1.
2. On the plan view, measure the distance between two points on the plan, and the height of the two same points on the profile. Triangulation Development step 2.
3. Calculate the true length of line AB using the formula L = (Lxy2 + Z2)½. Elsewhere, draw a circle of this radius. Point A will be at the center, and point B somewhere on the circumference of this circle. Triangulation Development step 3.
4. Repeat step 2 with two more points, one of which was used in step 2. In this case, I am using point B to point C. Triangulation Development step 4.
5. Calculate the true length of line BC as in step 3, and draw a circle of this radius at an arbitrary point on the radius of the circle drawn in step 3. As point B is at the center of this circle, connect the centers of the two circles with line AB. Triangulation Development step 5.
6. Now we complete the triangle by measuring the distance between points A and C. Triangulation Development step 6.
7. Calculate the true length and draw a circle of this radius around point A. Point C lies where the circles around points A and B intersect. Triangulation Development step 7.
8. Now measure one of the sides of an adjacent triangle. One of the ends of this triangle must be a point on the first triangle; ie. points A, B or C.H Here I am measuring between points C and the new D. Triangulation Development step 8.
9. Calculate the true length, draw a circle around point C. Point D will be on the circumference of this circle... but where? Triangulation Development step 9.
10. Measure the other side of the adjacent triangle. Triangulation Development step 10
11. Calculate the true length, and draw a circle around point B. The intersection of this circle and the one around point C is the location of point D. Triangulation Development step 11.
12. Repeat steps 8 through 11 until you've layed out the entire surface.

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© 2003 W. E. Johns