The short arc and its opposite
Convert both planets to absolute longitudes from 0° Aries, then travel along the shorter arc between them. Its center is one midpoint. The diametrically opposite longitude is the other end of the same midpoint axis. For a Sun at 20° Aries and a Moon at 10° Gemini, the longitudes are 20° and 70°. The short-arc center is 45°, or 15° Taurus; its opposite is 225°, or 15° Scorpio. Both belong to the axis at 45° modulo 180°.
Near the Aries boundary, ordinary arithmetic fails. A planet at 350° and one at 10° cluster around 0°; their short-arc center is 0°, not the naïve average of 180°. At exact opposition, two arcs are equally short. For 0° and 180°, the centers are 90° and 270°. The calculator labels that center ambiguous but still gives the single midpoint axis at 90° modulo 180°.
Why a 90° dial changes the order
A 90° dial places every longitude in a single quadrant by reducing it modulo 90°. Positions separated by 90°, 180° or 270° land together. For example, 15° Aries, 15° Cancer, 15° Libra and 15° Capricorn all fold to 15°. A 45° ordering reduces the longitude modulo 45° and groups even more angle classes. The folded coordinate is an index, not a new astronomical longitude: the chart point still has its original zodiac sign and degree.
A planet can be listed as a third-point contact to a midpoint when the folded angular distance is within an orb. The structure index writes this as focal ≈ A / B and lets you filter by focal point. Its four-point table tests two midpoint axes built from four distinct points; it excludes pairs that reuse a point. Both lists rank contacts by measured separation on the 90° circle. This calculator uses a user-selected 0.5° to 3° orb on the 90° dial and prints the measured distance. It excludes the pair’s two members when listing other contacts. A wider orb produces more rows; nothing in the arithmetic establishes an interpretation by itself.
Record quality and method boundaries
The local chart engine supplies apparent geocentric tropical positions for Sun through Pluto. If time and place are known, the Ascendant and Midheaven join the ten planets. If the birth time is unknown, local noon supplies an approximate planet set, but the angles are removed. The Moon travels quickly, so Moon-related midpoints can be especially sensitive to a missing hour. The site also asks for the historical civil UTC offset, which is not always today’s offset for that city.
A relationship midpoint composite combines two people’s corresponding planets, whereas this natal table pairs different points from one chart. Calculate your own midpoint table.