A family of hyperbolas associated to a triangle

In this note, we explore an apparently new one parameter family of conics associated to a triangle. Given a triangle we study ellipses whose one axis is parallel to one of sides of the triangle. The centers of these ellipses move along three hyperbolas, one for each side of the triangle. These hyperbolas intersect in four common points, which we identify as centers of incircle and the three excircles of the triangle. Thus they belong to a pencil of conics. We trace centers of all conics in the family and establish a surprising fact that they move along the excircle of the triangle. Even though our research is motivated by a problem in elementary geometry, its solution involves some non-trivial algebra and appeal to effective computational methods of algebraic geometry. Our work is illustrated by an animation in Geogebra and accompanied by a Singular file

NeRFlame: FLAME-based conditioning of NeRF for 3D face rendering

Traditional 3D face models are based on mesh representations with texture. One of the most important models is FLAME (Faces Learned with an Articulated Model and Expressions), which produces meshes of human faces that are fully controllable. Unfortunately, such models have problems with capturing geometric and appearance details. In contrast to mesh representation, the neural radiance field (NeRF) produces extremely sharp renders. But implicit methods are hard to animate and do not generalize well to unseen expressions. It is not trivial to effectively control NeRF models to obtain face manipulation. The present paper proposes a novel approach, named NeRFlame, which combines the strengths of both NeRF and FLAME methods. Our method enables high-quality rendering capabilities of NeRF while also offering complete control over the visual appearance, similar to FLAME. Unlike conventional NeRF-based architectures that utilize neural networks to model RGB colors and volume density, NeRFlame employs FLAME mesh as an explicit density volume. As a result, color values are non-zero only in the proximity of the FLAME mesh. This FLAME backbone is then integrated into the NeRF architecture to predict RGB colors, allowing NeRFlame to explicitly model volume density and implicitly model RGB colors