Declination and maximum altitude of the Sun above the horizon on a given date
This online calculator calculates the declination of the Sun on a given date and the maximum altitude above the horizon on that day for a given latitude
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A short text about the declination and altitude of the Sun above the horizon as a function of latitude can be found below the calculator.
The path of the Sun at different latitudes
Declination means declination in the second equatorial coordinate system, which, unlike declination in the first equatorial coordinate system, does not change due to the daily motion of the Earth. This is due to the fact that the second coordinate in the second equatorial coordinate system  direct ascent, is counted from the point of the vernal equinox, which is stationary. Accordingly, the Sun reaches its declination, and highest position, once a day, at true noon.
The maximum altitude of the Sun is related to the latitude at which the observer is located and the Sun's declination by the following relationship: h_{max} = 𝜎 + (90°  𝜑), where 𝜎 is the declination of the Sun, 𝜑 is the latitude of the place.
The maximum value of the Sun's declination 𝜎, on the day of the summer solstice, is equal to the angle of inclination of the Earth's axis  23°26′14″ (approximately, since the value of the angle of inclination is constantly changing slightly due to various effects). Accordingly, it can be seen that for latitudes smaller than 23°26′14″ (south of the Tropic of Cancer and north of the Tropic of Capricorn), the value in the formula above can sometimes exceed the maximum height of the Sun above the horizon of 90° degrees (zenith). In practice, this means that, in the case of the northern hemisphere, the Sun culminates not south of the zenith (when at noon the shadow points north), but north, and consequently the shadow at noon points south. The calculator always shows a value less than 90° by subtracting the resulting value from 180°.
On the Tropic of Cancer itself, on the day of the summer solstice, the sun passes through the zenith, that is, at noon it is directly overhead  the day of zero shadow. South of the Tropic of Cancer, the day of zero shadow happens twice a year  when the declination of the Sun is equal to the latitude of the place of observation. But for latitudes above 66°33′46″ (90°  23°26′14″) (north of the Arctic Circle and south of the Arctic Circle), the altitude can be negative, corresponding to a polar night.
To summarize all this, I will cite the astronomy characteristics of temperature belts from Bakulin's textbook^{1}, Chapter 1, § 17. The daily path of the Sun at different latitudes:
 In frigid zones (from 𝜑 = ± 66° 34' to 𝜑 = ± 90°), the Sun can be a nonsunsetting and nonsunrise luminary. Polar day and polar night can last from 24 hours to half a year.
 In temperate zones (from 𝜑 = ± 23° 26' to 𝜑 = ± 66° 34') the Sun rises and sets every day, but is never at zenith. Polar days and nights never occur here. The duration of day and night is shorter than 24 hours. In summer, the day is longer than the night and vice versa in winter.
 In the torrid zone (from 𝜑 = + 23° 26' to 𝜑 =  23° 26') the Sun is also always a rising and setting luminary and is at its zenith twice a year (once in the tropics) at noon. The days it happens depends on the latitude, and at the equator the Sun at zenith happens on the day of the vernal and the day of the autumnal equinoxes.

A course of general astronomy / P. I. Bakulin, E. V. Kononovich, V. I. Moroz.  Moscow: Nauka, 1976. ↩
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