## Using Spherical Coordinates

In this database, spherical coordinates are used to describe light/sensor directionality when collecting color reflectance spectra. This page provides a short refresher on how to utilize spherical coordinates.

Spherical coordinates have three parts: the radius, azimuthal angle, and polar angle, written as (ρ, θ, φ).

These parts are described in relation to three axes: the x-axis, y-axis, and z-axis. In the Figure 1 example we have oriented them so that the origin is in the center of the specimen; the x-axis lying on the specimen's line of bilateral symmetry, with the positive x-axis extending anteriorly from the origin; the positive y-axis extending right laterally from the origin; and the z-axis extends dorsally from the origin.

Figure 1. Imagining the specimen standing on a flat surface such as a table, the x-y plane would be parallel to the table top and the positive z-axis would be pointed straight up at the ceiling.

When imaging color patches, the origin point will be located at the color patch. The color patch axes must be parallel to the example axes from Figure 1.

Figure 2. The origin now lies on the blue color patch on the butterfly's wing.

The radius, azimuth angle, and polar angle describing the light/sensor directionality are relative to the origin point and axes of the color patch. Figure 3 shows how these would be found on the example butterfly from above.

The probe is elevated distance z from the xy plane parallel to the color spot, and along this xy plane is distance r from the color spot, i.e. the origin point.

Radius, ρ, is the distance from the light source/sensor to the color spot, and can otherwise be calculated as the square root of (z² + r²).

Azimuth angle, θ, is the angle between the positive x-axis and the line denoted by r.

Polar angle, φ, is the angle between the positive z-axis and the line from the light source/sensor to the origin.

Figure 3. The magnified view illustrates the arrangement of light sources and spectrometer sensor that would be found on a fiber optic reflection probe for spectrometers.

## Required Measurements

Database entries require two spectral measurements per color patch, at minimum: "straight on" (90°) and angled (45°).

The coordinates for the "straight on" view would be either (ρ, 0°, 0°) for a measurement of a color patch on the dorsal side of the specimen, or (ρ, 0°, 180°) for a measurement from the ventral side.

The coordinates for the angled view will vary based on the peak reflectance of the patch. Azimuth angle θ will be whichever of the following angles has greatest reflectance: 0°, 90°, 180°, or 270°. If all four angles have the same reflectance, use the data from the 0° orientation. Polar angle φ will be 45° for a measurement of a color patch on the dorsal side of the specimen or 135° for a measurement from the ventral side.

Spectral data should be accompanied by light photographs of the color patch in the same orientation as the spectral measurement. The radius ρ for light photographs does not need to be an exact measurement and an approximation will suffice.