![]() This cone has a solid angle of precisely one steradian (or one “solid radian”), which is abbreviated sr. Now, imagine a sphere with radius r, and with a cone-shaped section whose base has a surface area of r * r, or r 2 (FIG. Radians are more useful simply because they are related to the geometry of the circle rather than some magic number - they are easier to visualize and so understand. ![]() (The reason for the magic number 360 is lost in history, according to Wikipedia.) This means that one radian is equal to 360 / (2 * pi) = 180 / pi degrees, which is approximately 57.3 degrees. Most of us are used to thinking of angles in terms of degrees ñ there are 360 degrees in a circle. The resultant angle is precisely one radian, which is abbreviated rad. What this means is that if we take a piece of string with length r, we will need to stretch it by a factor of two pi (6.28328 …) to wrap around the circumference of the circle.īut suppose we wrap the string with length r part way around the circle (FIG. (Remember that 1980s-era four-function calculator ñ it is all you will need for this.) If you remember anything at all from mathematics in school, it is that the circumference C of a circle with radius r is equal to two times pi times its radius, or: So, we start by visualizing a circle (FIG. ![]() I learned this from a professor of mine whose specialty was hyperspace geometry - he could “easily imagine” four- and five-dimensional objects by mentally projecting them into three-dimensional shapes and imagining how their shadows changed as he rotated the objects in his mind. Just as lighting designers can look at architectural drawings and imagine lighting designs, mathematicians can look at a set of equations - which are really nothing more than an arcane written language - and visualize new mathematical concepts and proofs. The all-knowing Wikipedia has an answer: it is the measure of a “solid angle.” Going to the Wikipedia definition of this phrase, we see:īut now for a trade secret: most mathematicians do not think in terms of equations like these double integrals. A lumen is easy enough to understand, but what the blazes is a “steradian”? We measure the luminous intensity of a light source in candela, which is defined as “one lumen per steradian” (IES 2010). ![]() In studiously ignoring the mathematics of a topic, we all too often overlook the underlying concepts that help us better understand what we are interested in.Īn example from lighting design: luminous intensity. Our fear (note the implicit “we”) can, however, disadvantage us in subtle ways. As an electrical engineer in the 1980s for example, I never needed anything more than a four-function calculator to do my work designing billion-dollar transportation systems. I would tell you the exact numbers, but you would need to understand statistical analysis …įortunately, we can mostly muddle through our lives without having to deal with statistics, vector calculus, differential geometry, algebraic topology and all that. Eng., FIES, Senior Scientist, SunTracker Technologies Ltd.ĭo you suffer from math anxiety? A surprising number of us do (e.g., Wigfield 1988). ![]()
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