A comprehensive ray tracing study on the impact of solar reflections from glass curtain walls

Abstract

To facilitate the investigation of the impact of solar reflection from the façades of skyscrapers to surrounding environment, a comprehensive ray tracing model has been developed using the International Commerce Centre (ICC) in Hong Kong as an example. Taking into account the actual physical dimensions of buildings and meteorological data, the model simulates and traces the paths of solar reflections from ICC to the surrounding buildings, assessing the impact in terms of hit locations, light intensity and the hit time on each day throughout the year. Our analyses show that various design and architectural features of ICC have amplified the intensity of reflected solar rays and increased the hit rates of surrounding buildings. These factors include the high reflectivity of glass panels, their upward tilting angles, the concave profile of the ‘Dragon Tail’ (glass panels near the base), the particular location and orientation of ICC, as well as the immense height of ICC with its large reflective surfaces. The simulation results allow us to accurately map the date and time when the ray projections occur on each of the target buildings, rendering important information such as the number of converging (overlapping) projections, and the actual light intensity hitting each of the buildings at any given time. Comparisons with other skyscrapers such as Taipei 101 in Taiwan and 2-IFC (International Finance Centre) Hong Kong are made. Remedial actions for ICC and preventive measures are also discussed.

Appendices

Appendix A

Ray-plane intersection point detection

For any given planar polygon, its surface can be described as a subset of an infinite plane with the same unit normal vector n. For a given light ray with unit directional vector r, the coordinate of intersection between the ray and the plane is given by

$$ {P}_{\mathrm{intersect}}=\frac{\mathbf{n}\cdot \left(C-{r}_0\right)}{\mathbf{n}\cdot \mathbf{r}}\mathbf{r}+{r}_0 $$

(11)

where C is the coordinate of any point on the plane (such as one of the vertices of the polygon) and r ₀ is the starting coordinate of the light ray. If n ⋅ r = 0, the ray will never meet the plane. If $\frac{\mathbf{n}\cdot \left(C-{r}_0\right)}{\mathbf{n}\cdot \mathbf{r}}$ is positive, then the ray intersects with the plane in the forward direction; otherwise, if it is negative, the ray intersects with the plane in the reverse ray direction.

Polygon inclusion test

Successfully obtaining a ray-plane intersection point with (Eq. 11) does not imply the ray will intersect with a polygon on the plane. It is necessary to perform a second test to check whether the point falls within the actual polygon.

The polygon of interest is first subdivided into triangles, and the following checks based on Barycentric coordinates (Ericson 2004) will determine if a particular coordinate falls within any of the triangles. For a given triangle with vertices A, B and C, and a point P which we wanted to test for its inclusion in the triangle, we can define the following vectors (see Fig. 28):

$$ \begin{array}{lll}{v}_0=C-A,\hfill & {v}_1=B-A,\hfill & {v}_2=P-A\hfill \end{array} $$

Fig. 28

Triangle point inclusion test

The Barycentric coordinates (U, V) are given by

$$ d=\left({v}_0\cdot {v}_0\right)\left({v}_1\cdot {v}_1\right)-\left({v}_0\cdot {v}_1\right)\left({v}_0\cdot {v}_1\right) $$

$$ U=\left(\left({v}_0\cdot {v}_2\right)\left({v}_1\cdot {v}_1\right)-\left({v}_0\cdot {v}_1\right)\left({v}_1\cdot {v}_2\right)\right)/d $$

$$ V=\left(\left({v}_0\cdot {v}_0\right)\left({v}_1\cdot {v}_2\right)-\left({v}_0\cdot {v}_1\right)\left({v}_0\cdot {v}_2\right)\right)/d $$

If (U ≥ 0) and (V ≥ 0) and ((U + V) < 1) are satisfied, then P is within the triangle; otherwise, P is outside of the triangle.

Appendix B

To model the sun ray intensity on earth’s surface, we first derive the relationship between the altitude angle of the sun (θ) and the length of atmosphere that the rays have to travel through (L).

Figure 29 illustrates the geometric problem with L and θ and other relevant angles and dimensions

Fig. 29

Model of sun ray passing through the earth’s atmosphere

According to the Intersecting Chord Theorem,

$$ L\left(L+x\right)=a\left(a+2r\right) $$

Therefore, by completing the square,

$$ \begin{array}{l}{\left(L+\frac{x}{2}\right)}^2-\frac{x^2}{4}=a\left(a+2r\right)\hfill \\ {}L=-\frac{x}{2}\pm \sqrt{\frac{x^2}{4}+a\left(a+2r\right)}\hfill \end{array} $$

Also,

$$ \begin{array}{l}\begin{array}{ll}x=2r \sin \frac{\beta }{2},\hfill & \beta =180{}^{\circ}-2\phi =180{}^{\circ}-2\left(90{}^{\circ}-\theta \right)=2\theta \hfill \end{array}\hfill \\ {}\hfill \Rightarrow x=2r \sin \theta \hfill \\ {}\hfill \therefore L=-r \sin \theta \pm \sqrt{r^2{ \sin}^2\theta +a\left(a+2r\right)}\hfill \end{array} $$

For positive L

$$ L=\sqrt{r^2{ \sin}^2\theta +a\left(a+2r\right)}-r \sin \theta $$

(12)

According to Beer–Lambert law, the general expression for atmospheric light attenuation (Overington 1976) through distance L assuming negligible light refraction is given by

$$ I={I}_0 \exp \left(-CL\right) $$

(13)

where I ₀ is the initial solar intensity before entering earth’s atmosphere and C is the light extinction coefficient in the atmosphere (Overington 1976).

When θ = 90°, L is at its minimum (L = L _min = a), and the solar intensity I on earth’s surface is at its maximum (I _max). Therefore,

$$ {I}_{\max }={I}_0 \exp \left(-C{L}_{\min}\right)={I}_0 \exp \left(-Ca\right) $$

$$ \Rightarrow {I}_0=\frac{I_{\max }}{ \exp \left(-Ca\right)} $$

This allows the I ₀ in (Eq. 13) to be expressed in terms of I _Max yielding

$$ \begin{array}{rcl}I=\frac{I_{\max }}{ \exp \left(-Ca\right)} \exp \left(-CL\right)& =& {I}_{\max } \exp \left[-C\left(L-a\right)\right]\kern1em \end{array} $$

By substituting the definition of L from (Eq. 12), we get the final ray intensity model (Eq. 6):

$$ {I}_{\mathrm{model}}\approx {I}_{\max } \exp \left[-C\left(\sqrt{r^2{ \sin}^2\theta +a\left(a+2r\right)}-r \sin \theta -a\right)\right] $$

Appendix C

Full list of building names and locations in the model as depicted in Fig. 30:

ᅟ ᅟ

Fig. 30

Location of all 24 building estates (73 buildings) in the model