da Vinci bridges

Below the diagram are some notes on the calculations.

number of segments =
total span (dark green line) = mm (approx.)

For upright side-members, high == wide; for flat side-members, high == thick.

To calculate the geometry of the bridge, we view each side-member as consisting of five sections:

• a section under the cross-member (wide mm);
• two ends above the neighbouring cross-members, which don't affect the geometry so they don't have a particular length;
• two sections between cross-members (reach mm).

        long == wide + 2*reach; reach =  mm

The angle between members is calculated from the right-angled triangle formed by:

• the angle we want is at the point where the side-members cross, halfway along the reach, marked with a blue circle;
• the point on the centre-line of the side-member under the edge of the cross-member, marked with a blue right-angle symbol;
• the point halfway up the edge of the cross-member, where the blue line meets it in the lower close-up diagram.

        tan(theta/2) = (high/2 + thick/2) / (reach/2)
	theta =  degrees

This is an approximation. It fails when the cross-members are very thick, because I estimated that halfway up the cross-member is close enough to halfway between the side-members. This is not quite true because the edge of the cross-member does not form the same angle with both side-members. I also guesstimated that the crossing point is the centre of the reach.

The angle theta is also the angle between each segment of the bridge, shown by the red lines that join at the centre of curvature. This gives us the number of segments to maximize the span of the bridge:

        segs = floor(180 / theta)

The total span is another approximation. We calculate an outer radius as the distance from the centre of curvature to a point where the ends of two side-members would cross above a cross-member (if their ends extend far enough). We estimate that these outer crossing points close enough to the ends of the side-members, so we can use them to locate the ends of the entire bridge. The outer radius is calculated from the inner radius, from the centre of curvature to the centre of a side-member.

        tan(theta/2) = (reach/2 + wide/2) / inner
        cos(theta) = inner / outer

PS. If this page has a blank space, you need to enable JavaScript so that it can draw the interactive diagram.

Tony Finch <dot@dotat.at>