We present an optical surface in closed form that can be used to design lenses for controlling relative illumination on a target surface. The optical surface is constructed by rotation of the pedal curve to the ellipse about its minor axis. Three renditions of the surface are provided, namely as an expansion of a base surface, and as combinations of several base surfaces. Examples of the performance of the surfaces are presented for the case of a point light source. (C) 2016 Society of Photo-Optical Instrumentation Engineers (SPIE)

The volume of lava flows provide important information on the magnitude of volcanic eruptions, and accurate volumes are necessary to produce reliable models of lava flow emplacement or constrain the internal structure of volcanoes. The most accurate lava flow volumes are obtainable when the topography before and after an eruption are both known, but information for the topography before lava flow emplacement is absent in non-historic lava flows. To calculate the volume of non-historic lava flows, this pre-emplacement topography needs to be reconstructed. Common methods for this include using inverse distance-weighted averages or global polynomial interpolation methods, but these can still underestimate the volume of the flow, and the surface of the flow itself is not considered in these interpolations. A new calculation method seems necessary to better constrain the volume of lava flows, and including the lava flow surface in the volume calculation, given that it is generally excluded during interpolation of pre-emplacement topography, may be the solution to improving lava flow volume calculation for flows where the base surface is unknown. The 2012-2013 Tolbachik lava flow is used to look at potential relationships due to the availability of elevation data before and after the eruption. A quantitative analysis on the relationships between the slope of topography before and after lava flow emplacement and on the relationship between the slope and thickness of lava flows is performed. In addition to this, the slope of the topography calculated over local and regional scales is used as a new interpolation method, and the calculated thickness from the interpolated surface is compared to the known thickness for the lava flow.