Hemisphere #

A hemisphere describes the subdivision of a dome into discrete patches. Each patch has an elevation and an azimuthal domain that describes its angular extent in the hemisphere and its area in steradian.

Construct Hemisphere#

Cell Count#

The target number of cells the dome should be subdivided into. Different subdivision types allow for different amount of cells and this value will be rounded down to the closes possible cell count $C(x)$ for the specific subdivision.

Info

The cell count affects the accuracy of analyses but also the time needed to run them.

Subdivision#

The dome subdivision type. Different subdivisions use different equations to subdivide the dome into discrete patches. All subdivisions are symmetric over the y-axis and have horizontal bands of patches with an inflection point at zenith.

Equal Angle#

Subdivides the hemisphere into cells of identical elevation and azimuthal angle domains resulting in equal amount of cells in each band and a higher density of cells at zenith. The cell count $C(x)$ is bound by two rules. There has to be an equal amount of cells in each quadrant of the hemisphere and the amount of cells in each quadrant has to be a perfect square. $$ C = 4 \cdot x^2 \quad \text{where } x \in \mathbb{N} $$

Equal Area Aspect#

The equal area aspect subdivision tries to split the dome into patches with equal area and aspect ratio. The cells at zenith can be marginally different in size to the rest of the cells in the hemisphere but this is currently the most accurate way to get an even distribution of cells across the dome. There is no limitation to the cell count for this subdivision, it will always produce the amount of cells specified.

Equal Area Congruent#

The equal area congruent subdivision aims at creating equal area cells while making the subdivision congruent, meaning equal amount of cells in each band. The resulting cells will have equal area but significantly varying aspect ratios. The cell count $C(x)$ is bound by two rules. There has to be an equal amount of cells in each half of the hemisphere and the amount of cells in each half has to be a perfect square. $$ C = 2 \cdot x^2 \quad \text{where } x \in \mathbb{N} $$

Equal Area Congurent Subdivision — A hemisphere subdivided with an equal area congruent subdivision into 512 patches.

Reinhart Tregenza#

The Reinhart Tregenza subdivision is the standard when subdividing a dome for environmental analysis. It is based on a 145 cell subdivision with 7 bands of 12° zenith angle plus a 12° diameter circular patch at zenith. The cell count $C(x)$ is limited to a further subdivision of the initial 145 patches. $$ C = 144 \cdot x^2 + \lceil \frac{x^2}{2} \rceil \quad \text{where } x \in \mathbb{N} $$

Hemisphere Mesh#

The hemisphere can be represented as a mesh. This can be useful for plotting hemisphere related data or for other geometric operations outside of Cyclops.

Projection#

The projection decides how the subdivided hemisphere should be represented. Turning the spherical $(\varphi, \theta)$ coordinates (with $\varphi$ the altitude angle, $0 \leq \varphi \leq \pi/2$, and $\theta$ the azimuth, $0 \leq \theta \leq 2\pi$) into Cartesian coordinates $(x, y, z)$ on the sphere or $(X, Y)$ on the plane.

Cartesian#

The cartesian projection unrolls the dome into a cylinder and then unrolls the cylinder into a planar region of $2\pi$ width and $\pi/2$ height. This gives a direct translation between the spherical and cartesian coordinates. $$ X = \theta , \quad Y = \varphi $$

Stereo#

The stereo or stereographic projection transforms the hemisphere into a unit disc on the $(X, Y)$ plane. It is conformal meaning that it preserves angles locally and minimise shape distortion. $$ X = \frac{cos(\varphi)}{1 + cos(\varphi)} \cdot cos(\theta) $$ $$ Y = \frac{cos(\varphi)}{1 + cos(\varphi)} \cdot sin(\theta) $$

Equidistant#

The equidistant projection, also known as azimuthal equidistant projection transforms the hemisphere into a unit disc on the $(X, Y)$ plane. The radius on the disc has a linear relationship with $\varphi$, preserving the distance from the centre point to any other point on the disk. $$ X = \frac{(\pi/2 - \varphi)}{\pi/2} \cdot cos(\theta) $$ $$ Y = \frac{(\pi/2 - \varphi)}{\pi/2} \cdot sin(\theta) $$

Spherical#

The spherical projection is the only 3-dimensional projection in Cyclops. It translates the spherical $(\varphi, \theta)$ coordinates to into Cartesian coordinates $(x, y, z)$ on a unit dome. $$ x = cos(\varphi) \cdot cos(\theta) $$ $$ y = cos(\varphi) \cdot sin(\theta) $$ $$ z = cos(\varphi) $$

Beckers, Benoit & Beckers, Pierre. (2012). A general rule for disk and hemisphere partition into equal-area cells. Computational Geometry. 45. 275–283. 10.1016/j.comgeo.2012.01.011.

Tregenza, Peter. (1987). Subdivision of the sky hemisphere for luminance measurements. Lighting Research & Technology - LIGHTING RES TECHNOL. 19. 13-14. 10.1177/096032718701900103.