Solar altitude (or elevation, α)

Angle between a) a line segment originating from the observer to the Sun, and b) the line segment defined by the horizontal projection of the first segment onto the horizontal plane.

“Measure of the angle formed between the horizontal projection of the ray and the ray itself. In other words, the measure of the angle formed between the horizontal plane and the line segment from the observer to the apparent Sun. Given a vertical plane containing the observer (O), a point on the horizon (H), and the Sun (S), it is the angle SÔH formed by segments OS and OH. Its value ranges from 0° to 90°.” (FERNANDES, 2023)

Figure 1. Solar altitude and its representation via concentric circles on the horizontal plane. (FERNANDES 2023).

Julian year

A period of 365.25 days (which includes the 0.25 day fraction to account for leap years in the Julian calendar).

Solar azimuth (γ or φ, depending on the convention)

Angle on the horizontal plane, measured from a reference direction (usually North or South).

Solar azimuth (AZI) is the “Measure of the angle between the observer-North line and the ray's projection on the horizontal plane (Figure 4, left). Given the horizontal plane containing the observer (O), True North (N), and the horizontal projection of the Sun (S’), AZI is the angle NÔS’ formed between segments ON and OS’. Its value increases clockwise, ranging from 0° to 360°. On the sun chart, rays indicating the azimuths are plotted every 10°”. (FERNANDES, 2023)

Figure 2. Solar azimuth AZI (left). Azimuth indicator lines, every 10° (center). Cartesian X and Y coordinates of the Sun's projection on the horizontal plane (right).

Julian Calendar

Implemented by Julius Caesar in 46 BC, it was the predecessor to the current Gregorian calendar and introduced the concept of leap years every four years.

In technical and astronomical contexts, it is often associated with the concept of the Julian Day, a continuous count of days that facilitates precise calculation of the Earth's orbital position and, consequently, solar declination.

Sun chart

Diagram representing the apparent sun paths on the celestial dome. In azimuthal projections, centered on the observer, sun charts represent the paths of the Sun and the solar altitudes and azimuths along these paths.

In the case of charts generated in Antropo.Sol, the sun chart can be two-dimensional (2D), projected on the horizontal plane, or three-dimensional, projected on the celestial dome (3D).

Associated terms: sunchart, sun chart, sunpath, sun path, sunpath2d, sunpath3d, 3D Sun-Path.

Figure 3. Illustration of Vitruvius' Method (Vitruvio 1914) to obtain the side view of the apparent sun paths' projection on the celestial dome around an observer at the approximate latitude of Rome, 41.6° (or) and an outline of the sun chart deduced by (FERNANDES 2023) for the same location. Source: adapted from Vitruviu's (Vitruvio 1914) and supplemented by (FERNANDES 2023).

Figure 4. Two-dimensional sun chart (2D sun chart). Orthographic projection. Location: Curitiba, PR. Source: elaborated in AntropoSol.Web.

Figure 5. Three-dimensional sun chart (3D sun chart). Orthographic projection. Location: Curitiba, PR. Source: elaborated in AntropoSol.Web.

Circles of equal solar altitude

Horizontal projection of the circular line of points with the same solar altitude (center). Generally, sun charts display the projection of several circular lines of equal solar altitude spaced every 10°. (FERNANDES, 2023)

Figure 6. Imaginary topocentric system, centered on the observer (left). Solar altitude and horizontal projection of a line of points with the same solar altitude (center). Circular lines of equal solar altitude, spaced every 10° (right). FERNANDES, 2023.

Celestial dome or hemisphere

This is the imaginary hemisphere (also called celestial vault), situated above the observer's plane, with an infinite radius, upon which the celestial bodies are imagined to move. It corresponds to the upper half of the celestial sphere centered on the observer.

Figure 7. Celestial dome or hemisphere. Source: (FERNANDES (2023).

Cartesian coordinates

Cartesian coordinates are informational elements of a system, bridging algebra and geometry, which uses numerical values to specify the exact position of a point on a plane or in space.

Figure 8. Cartesian X and Y coordinates of the horizontal projection of point S. Source: (FERNANDES 2023).

Ecliptic coordinates

The ecliptic coordinate system is a spherical celestial coordinate system. It uses the ecliptic as its reference plane and its zero point is the vernal equinox.

Equatorial coordinates

The equatorial coordinate system is a positioning system that locates the Sun on the celestial sphere by referencing the Earth's equator projected onto the celestial sphere, creating the celestial equator, and the Earth's axis of rotation. It differs from horizontal coordinates because it does not move as the observer shifts on the Earth's surface. Equatorial coordinates describe the Sun's position relative to the center of the Earth rather than the observer's position.

Solar declination (DEC)

Solar declination (DEC) is the angle formed between the Earth's equatorial plane and the Earth-Sun line.

Figure 9. Declination: angle formed between the Earth's equatorial plane (gray sphere) and the Earth-Sun line (the Sun shown as the white sphere, at the center of the system). (FERNANDES 2023).

Day of the year (N)

A value between 0 and 365, corresponding to the day of the year in a continuous sequence.

Day of the month

A value between 0 and 31, corresponding to the day of the month in a continuous sequence.

Julian day

The term “Julian day” is used to indicate the number of a day in a continuous series, without separating it into years, months, or weeks. The zero point (0.0) of this series corresponds to noon on January 1, 4713 BC, in the Julian calendar.

Ecliptic

The apparent path of the Sun projected on the celestial sphere throughout the year.

Equation of Time

Used to calculate the difference, in minutes, between true solar time (based on the actual position of the Sun on the celestial dome) and mean solar time (used to keep clock times uniform). The difference calculated occurs due to the eccentricity of the Earth's orbit and the tilt of the Earth's axis.

Associated terms: Equation of Time.

Equinox

The word equinox comes from Latin and means “equal nights” – when the Sun is at the equinox, the length of the night is the same across the entire planet.

Celestial sphere

An imaginary surface of spherical shape, with a radius of one (infinite), on which the apparent movements of the Sun occur. It is divided into the day hemisphere and the night hemisphere. The highest point of the day hemisphere is called the Zenith and the lowest point of the night hemisphere is called the Nadir (Fernandes, 2023).

Figure 10. Imaginary topocentric system, centered on the observer (left). Solar altitude and horizontal projection of a line of points with the same solar altitude (center). Circular lines of equal solar altitude, spaced every 10° (right). (FERNANDES 2023.

Time zone

A region of the Earth's surface located between two half-planes that touch at the planet's axis of rotation and form a 15° angle between them. In total, there are 24 time zones, meaning 24 spindle-shaped regions on the planet's surface.

Solar geometry

A field of knowledge regarding the study and prediction of the Sun's apparent paths and the design of devices to capture, block, or control the incidence of direct solar radiation.

Civil time (clock time)

Clock time or civil time is the reference used for daily activities, such as the time to wake up, have breakfast, start work, or schedule a gathering with friends or family.

Apparent solar time

Apparent solar time is related to the Sun's position on the celestial dome and can be obtained with a sundial. In solar time, at noon, the Sun will be approximately on the South-North line. For locations that are not on the central meridian of their respective time zone, there is a divergence between apparent solar time and civil time. This difference increases as the observer moves further away from the central meridian of the time zone.

Latitude

It is the geographic coordinate, in degrees, indicating the angular distance, measured along the meridian, of a point on the Earth's surface relative to the Equator. It varies from 0° at the Equator to 90° at the North (positive) or South (negative) poles, being the determining factor for the angle at which solar rays strike a surface throughout the seasons.

Horizon line

A circular line indicating the intersection between the celestial sphere and the observer's plane. (FERNANDES 2023.

Longitude

Refers to the angular distance of a point relative to the Prime Meridian (Greenwich), measured on the equatorial plane. Its values range from 0° to 180°, to the East or West, and its main utility in solar geometry is calculating the 'true solar time', allowing for the adjustment of the difference between clock time and the Sun's actual position.

Shading mask

A diagram upon which lines are drawn indicating the moments when a shadow covers a certain extent of the opening in question. Usually, the lines where the shadow covers 50% and/or 100% of the opening are highlighted.

Associated terms: ShadowMap, Shadows, Sun Shadows, ShadeMap.

Meridian

It is any imaginary great semi-circle that connects the geographic North and South poles, intersecting the Equator perpendicularly. All points located on the same meridian share the same solar noon, serving as the reference line for measuring longitude and orienting facades relative to cardinal points.

Nadir

It is the point directly below the observer on the celestial sphere. That is, the lowest point of the night hemisphere.

Figure 11. Imaginary topocentric system, centered on the observer and with the Zenith immediately above the observer. (FERNANDES 2023.

Observer (O)

The center of the system, from where the apparent movements of the Sun are imagined to be viewed.

Observer's plane (HP)

Horizontal projection plane (HP) where the observer is imagined to be. In these cases, the HP is represented by a circular segment of the horizontal plane. (FERNANDES 2023.

Celestial poles

The celestial poles are the projections of the North Pole and South Pole onto the celestial sphere. The Zenith and Nadir will only coincide with the North Celestial Pole and the South Celestial Pole when the observer is positioned at the Geographic North Pole (Fernandes, 2023).

Vernal equinox point

The point defined by the intersection of the celestial equator's plane and the ecliptic plane, at the moment the Sun crosses the celestial equator from South to North. When the Sun is at this point, it marks the beginning of autumn in the Southern Hemisphere and spring in the North (Lima Neto, 2022).

Horizontal projection of the Sun (S’)

Projection of the Sun on the horizontal projection plane.

Sun's position on the celestial dome

The position of the Sun in the sky for an observer situated on the Earth's surface. In the fields of architecture, engineering, and geography, it is generally given by spherical coordinates, more specifically by the horizontal (or altazimuth) system. In it, the Sun's position is defined by two angles: Solar altitude and Solar azimuth. In this case, the length of the Observer-Sun ray is irrelevant (the Sun is considered to be on the celestial sphere).

Associated terms: SunPosition, Sun Position.

Sunshade

A building element designed to block or regulate the incidence of direct solar radiation. Usually, sunshades are specific elements that differentiate and stand out in the building's geometry. However, other elements can fulfill the primary function of sunshades, such as canopies, eaves, awnings, and overhangs extending beyond the main body of the building.

Associated terms: Brise, Brise-soleil, Brises-soleil, Solar shading device, External shading device.

Horizontal sunshade

Figure 12. Horizontal sunshade. Source: elaborated in AntropoSol.Web.

Vertical sunshade

Figure 13. Vertical sunshade applied to the right side of the window (right for someone looking from the inside out of the room). Floor plan. Source: elaborated in AntropoSol.Web.

Figure 14. Vertical sunshade applied to the right side of the window (right for someone looking from the inside out of the room). Three-dimensional view. Source: elaborated in AntropoSol.Web.

Finite sunshade

A finite sunshade is one whose dimensions do not significantly exceed the dimensions of the opening.

Infinite sunshade

An infinite sunshade is one whose dimensions go well beyond the dimensions of the opening. They can extend across an entire facade or gable.

Line of sight or Ray (Ray)

Line segment connecting the observer (O) to the observed object (S). In this case, the observed object is the Sun (Fernandes, 2023).

Julian century

A unit of time used to measure long periods. A Julian century corresponds to 36,525 days, which is 100 Julian years.

Azimuthal projection system

AntropoSol uses sun charts created according to azimuthal projections, more specifically orthographic, stereographic, and equidistant projections. Azimuthal projections of the sky are those in which solar altitudes are represented by concentric circles and azimuths are straight lines that intersect at the center of these circles.

These projections resemble the azimuthal projections of the Earth's surface described by Sossa and Korol (2017), in which the solar altitude lines are replaced by latitude lines and the azimuth lines are replaced by meridian lines.

Figure 15. General schematic of the azimuthal projection system. (FERNANDES 2023)

Azimuthal projection system: subtypes

The most recommended azimuthal systems for producing sun charts are: orthographic, equidistant, and stereographic. The gnomonic projection is widely used for producing sundials.

“Regarding the result, more specifically the projection of solar altitudes, the equidistant system offers a middle ground between the orthographic and stereographic projections (Figure 16). In the orthographic projection, near the edge, the distances between different projected solar altitudes are greatly reduced. In the stereographic projection, they are greatly expanded. Meanwhile, in the equidistant projection, the circles relating to solar altitudes are spaced uniformly.” (FERNANDES 2023)

Figure 16. Orthographic (left), equidistant (center), and stereographic (right) projections. Source: (FERNANDES 2023).

Equidistant projection system

This is a graphical representation method that projects the three-dimensional celestial dome onto a two-dimensional plane, maintaining a linear relationship between the solar altitude and the diagram's radius. In other words, it ensures that, in the representation, the circles relating to solar altitudes are equidistant from each other.

“Currently, there is no available method to operationalize the equidistant projection via descriptive geometry graphically. The projections are performed using trigonometry. But, resorting to descriptive geometry to illustrate this method didactically, the projection of solar altitudes can be thought of in two ways: a) projection onto a horizontal plane situated at Zenith height, above the observer (left figure); b) projection onto a horizontal plane situated at the observer's height (center figure).” (FERNANDES 2023)

Figure 17. Equidistant projection of solar altitudes onto a horizontal plane situated at Zenith height (left); Equidistant projection of solar altitudes onto a horizontal plane situated at the observer's height (center). c) Relationship between: 1) the difference between the 90° angle (Zenith) and 2) the solar altitude (ALT) versus the relationship between RADIUS’ and RADIUS in the equidistant projection (right). Source: (FERNANDES 2023).

“(...) according to Szokolay, sun charts using the equidistant projection were developed in 1948 (Szokolay, 2008). This date is quite recent and should not be taken as definitive, because the geometry involved in equidistant projection, although sophisticated for laypeople, is relatively simple for those familiar with trigonometry. As far as the author has been able to verify in the bibliography, contrary to what Szokolay points out, the first to use the equidistant projection system to represent the sky were the ancient Egyptians. Later, the projection system also appears in 11th-century Muslim textbooks and is described in mathematical terms in the 16th century (Sossa; Korol, 2017). In the 1963 edition of Design with Climate, Olgyay presented and helped popularize equidistant sun charts (Olgyay, 2008). According to Szokolay (2007), the equidistant projection is the most used in sun charts in the United States.

Some authors, such as Szokolay (2007), argue that equidistant representation does not characterize a projection method and would merely be a coordinate system. This argument is probably due to the lack of a structured system for performing the equidistant projection using descriptive geometry. Therefore, analytical geometry, more specifically trigonometry, is used. In the present author's opinion, Szokolay's assertion is incorrect. The equidistant projection does indeed constitute a projection system, as it has all the necessary elements, even though the position of the projective center is variable.” (Fernandes, 2023)

Associated terms: linear polar projection or azimuthal equidistant.

Figure 18. Example of an equidistant sun chart for the city of Curitiba. Source: AntropoSol.

Stereographic projection system

In the stereographic projection of the celestial dome, two positions can be imagined for the horizontal projection plane: a) at Zenith height (left); or b) at the observer's feet (Figure 19, center). In the stereographic projection, the value of the horizontal projection of the RADIUS will always be smaller than the value in the orthographic projection (Figure 19, right) (FERNANDES, 2023).

Figure 19. Stereographic projection of solar altitudes: projection onto a horizontal plane situated at Zenith height, above the observer (left); projection onto a horizontal plane situated at the observer's height (center). Stereographic projection of the observer-Sun distance (RADIUS’) (right). Source: (FERNANDES 2023).

“According to Sossa and Korol (SOSSA 2017), the stereographic projection of the sky was proposed in the 2nd century BC by the Greek astronomer Hipparchus (190 – 120 BC) and can be understood as a consequence of the gnomonic projection. According to Szokolay (2007, p. 10), it is the most widely used system for representing sun charts.” (FERNANDES 2023)

Figure 20. Example of a stereographic sun chart for the city of Curitiba. Source: AntropoSol.

Gnomonic projection system

The gnomonic projection was developed and systematized by the Greek philosopher Thales of Miletus (624–546 BC) in approximately 580 BC (Sossa; Korol, 2017). It was applied in astronomical observation, navigation, and cartography.

Orthographic projection system

“The orthographic projection is the most commonly used projection method in technical drawing and computer software. In it, the Sun's positions on the celestial dome are projected perpendicularly onto the horizontal projection plane.” (FERNANDES, 2023)

Figure 21. Orthographic projection of the circles indicating solar altitudes (left). Reduced projection of the observer-Sun distance (RADIUS’), two-dimensional representation (center). Reduced projection of the observer-Sun distance (RADIUS’), three-dimensional representation (right) (FERNANDES 2023).

Figure 22. Example of an orthographic sun chart for the city of Curitiba. Source: AntropoSol.

Zenith

It is the highest point of the celestial sphere, positioned exactly above the observer. That is, the highest point of the day hemisphere.

Figure 23. Imaginary topocentric system, centered on the observer and with the Zenith immediately above the observer (FERNANDES, 2023)

REFERENCES

• (FERNANDES 2023): FERNANDES, L. C. Proposta de uma ferramenta computacional para traçar cartas solares e dimensionar dispositivos para sombreamento. PARC Pesquisa em Arquitetura e Construção , Campinas, SP, v. 14, p. e023028, 2023. DOI: https://doi.org/10.20396/parc.v14i00.8671992
• LIMA NETO, G. B. Astronomia de Posição . São Paulo: Instituto de Astronomia, Geofísica e Ciências Atmosféricas (IAG), 2022.
• (SOSSA 2017) SOSSA, Rostislav; KOROL, Pavel. Historical Aspects of Development of the Theory of Azimuthal Map Projections. Studia Geohistorica , n. 3, p. 187, 2017.
• SZOKOLAY, Steven. Solar geometry . 2nd. ed. Brisbane: The University of Queensland, 2007. (PLEA notes, note 1).
• (VITRUVIO 1914). Vitruvio, The Ten Books on Architecture . London: Harvard University Press, 1914. 366 p.