Universal Sundial Equations

BACK Summary/Comparison of Inclined/Declined Perpendicular & Polar Gnomon and Bifilar Sundial Formulas. These formulas may be used for plug-n-chug calculations when designing a polar/perpendicular gnomon or bifilar sundial (the dial plane may have any orientation and be located anywhere in the world). All you need to calculate is the sun's altitude and azimuth for the dial's location (latitude and longitude). Most of the variables are defined at the bottom of the page (some are defined where they are introduced). NOTE: for a horizontal dial, the dial inclination of 0° results in division by zero; however, these formulas can still be used. Simply use a very small value for the inclination (e.g. inclination = 0.001°). (A close inspection of the formulas using the original inputs of solar azimuth & altitude, and dial inclination & declination reveals why this operation is possible.) Please refer questions/comments to: concretepantyhose-AT-gmail-DOT-com
Dial Type	DPx X-coordinate in the plane of the dial.	DPy Y-coordinate in the plane of the dial.
Bifilar The origin is the position on the dial plane directly beneath the crossed threads. bifilar sundial calculator derivation of bifilar formulas
Perpendicular Gnomon The origin is the position at which the gnomon attaches to the dial plane. polar/perpendicular gnomon sundial calculator derivation of polar/perpendicular gnomon formulas
Polar Gnomon The origin is the position at which the style intersects the dial plane. The quantity σ is the dot product between the dial plane unit normal, n, and the nodus' position vector, g. Note that gx = 0, gy = ±Gcos(lat) and gz = ±Gsin(lat). The appropriate sign must be chosen in accordance with the procedure outlined in the above referenced article that discusses the derivation of the formulas.
Figure 1. 3D Coordinate System Variables and Definitions The dial plane X-axis is a horizontal line through the origin. The dial plane Y-axis is perpendicular to the X-axis and also through the origin. G = length of gnomon I = dial plane inclination OH = horizontal thread offset (height). (The horizontal thread is parallel to the X-axis.) OV = vertical thread offset (height). (The vertical thread is parallel to the Y-axis.) (α, β, γ) = direction cosines of the sun's position x-dir : α = cos(alt) sin(azm) y-dir : β = cos(alt) cos(azm) z-dir : γ = sin(alt) where, alt = solar altitude and azm = solar azimuth (A, B, C) = direction cosines of the dial plane's unit normal vector (also of the perpendicular gnomon ) x-direction : A = -sin(Ι) sin(Δ) y-direction : B = -sin(Ι) cos(Δ) z-direction : C = cos(Ι) where Ι = the inclination of the dial plane and Δ = the declination of the dial plane The quantity ψ is the dot product between the solar position unit vector and the dial plane unit normal. The 3D coordinate system (in which the direction cosines are measured) is shown in Figure 1. The X-Y plane represents the horizontal.