Determine focal pin distances, string trammel lengths, perimeter, and area for architectural arches and oval framing.
Consider an architectural elliptical arch opening with a major axis width 2a of 40.0 inches and a minor axis height 2b of 24.0 inches:
Verification Note: At the top vertex (0, 12), the distance to each focal pin (±16, 0) is √(16² + 12²) = √(256 + 144) = √400 = 20.00 in; sum of focal distances = 20 + 20 = 40.00 in (exact 2a match).
Focal Distance c = sqrt(a^2 - b^2), where a is the semi-major axis (width / 2) and b is the semi-minor axis (height / 2).
Place two pins at focal points F1 and F2 (+/- c from center). Tie a non-stretch string loop around both pins so the total perimeter loop length equals 2a + 2c (or string span between pins = 2a). Pull taut with a pencil and scribe.
A trammel of Archimedes uses a rigid beam with two sliding pins running in perpendicular T-slots. A pencil attached to the beam traces an exact ellipse without string stretch.
Area = π * a * b. For a 40 in by 24 in ellipse (a=20 in, b=12 in): Area = 3.14159 * 20 * 12 = 753.98 sq inches (5.24 sq ft).
Perimeter ≈ π(a + b)[1 + 3h / (10 + sqrt(4 - 3h))], where h = (a - b)^2 / (a + b)^2. It provides accuracy within 0.005%.
Semi-axes: a = 24 in, b = 15 in. c = sqrt(24^2 - 15^2) = sqrt(576 - 225) = sqrt(351) = 18.73 inches from center.
Standard nylon or cotton string stretches under tension, distorting the ellipse into a bulging oblong shape. Use braided non-stretch wire or monofilament line.
An elliptical arch uses half an ellipse (the upper semi-ellipse) to span door, window, or fireplace openings, offering a graceful shallow rise.
Mark points A and B on a stiff paper strip equal to semi-major and semi-minor lengths. Align A on the minor axis and B on the major axis to mark perimeter points.
Yes, if height > width, the vertical axis becomes the major axis (a) and focal points shift to the vertical center line.
Eccentricity e = c / a. A circle has e = 0. Flatter, more elongated ellipses approach e = 1.
Perpendicular tangent lines at any point P on the ellipse bisect the angle formed by lines drawn from P to both focal points F1 and F2.
Multiply axis dimensions by 25.4 mm/inch. A 40x24 in ellipse equals 1016x609.6 mm (a=508 mm, b=304.8 mm, c=406.4 mm).
Use steel layout pins, non-stretch stainless wire, a laser level for axis squareness, and a flexible steel rule.
A 4-center arch approximates an ellipse using 4 circular arcs struck from 4 centers, allowing layout with standard compasses.
Layout radial elliptical ribs using a trammel beam, cut plywood ribs, and install flexible drywall over the elliptical skeletal frame.
Wear safety glasses, secure workpiece templates on sawhorses, and wear steel-toe boots when handling heavy curved arch templates.
Measure diagonal check dimensions across opposite quadrants; equal diagonal spans confirm true 4-quadrant symmetry.