Fundamental problem of heat transfer within a half-space due to a moving heat source of hyperelliptical geometry is studied in this work. The considered hyperelliptical geometry family covers a wide range of heat source shapes, including star-shaped, rhombic, elliptical, rectangular with round corners, rectangular, circular, and square. The effects of the heat source speed, aspect ratio, corners, and orientation are investigated using the general solution of a moving point source on a half-space and superposition. Selecting the square root of the heat source area as the characteristics length scale, it is shown that the maximum temperature within the half-space is a function of the heat source speed (Peclet number) and its aspect ratio. It is observed that the details of the exact heat source shape have negligible effect on the maximum temperature within the half-space. New general compact relationships are introduced that can predict the maximum temperature within the half-space with reasonable accuracy. The validity of the suggested relationships is examined by available experimental and numerical data for the grinding process, for medium Peclet numbers. For ultrafast heat sources, an independent experimental study is performed using a commercial CO2 laser system. The measured depth of the engraved grooves is successfully predicted by the proposed relationships.
