Bilinear Patch Research Articles

Geometric Modeling of Umbrella Surfaces

In this study, we present a novel method for modeling the canopy surface of an umbrella. Our approach involves representing the area between the ribs on the canopy as a trimmed bilinear patch. Furthermore, we conduct an in-depth exploration of various differential geometric properties of the umbrella surface. We introduce a method for unfolding the canopy surface onto a plane, which serves as a valuable technique for fabricating a cardboard template to accurately cut canopy fabrics. To validate the effectiveness of our geometric modeling method, we apply it to several umbrella models, showcasing its practical application and benefits.

Ray/Ribbon Intersections

We present a new ray tracing primitive---a curved ribbon, which is embedded inside a ruled surface. We describe two such surfaces. Ribbons inside doubly ruled bilinear patches can be intersected by solving a quadratic equation. We also consider a singly ruled surface with a directrix defined by a quadratic Bézier curve and a generator---by two linearly interpolated bitangent vectors. Intersecting such a surface requires solving a cubic equation, but it provides more fine-tuned control of the ribbon shape. These two primitives are smooth, composable, and allow fast non-iterative intersections. These are the first primitives that possess all such properties simultaneously.

Fast analytical motion blur with transparency

We introduce a practical parallel technique to achieve real-time motion blur for textured and semi-transparent triangles with high accuracy using modern commodity GPUs. In our approach, moving triangles are represented as prisms. Each prism is bounded by the initial and final position of the triangle during one animation frame and three bilinear patches on the sides. Each prism covers a number of pixels for a certain amount of time according to its trajectory on the screen. We efficiently find, store and sort the list of prisms covering each pixel including the amount of time the pixel is covered by each prism. This information, together with the color, texture, normal, and transparency of the pixel, is used to resolve its final color. We demonstrate the performance, scalability, and generality of our approach in a number of test scenarios, showing that it achieves a visual quality practically indistinguishable from the ground truth in a matter of just a few milliseconds, including rendering of textured and transparent objects. A supplementary video has been made available online.11Supplementary video available here

Practical Product Sampling by Fitting and Composing Warps

AbstractWe introduce a Monte Carlo importance sampling method for integrands composed of products and show its application to rendering where direct sampling of the product is often difficult. Our method is based on warp functions that operate on the primary samples in [0,1)n, where each warp approximates sampling a single factor of the product distribution. Our key insight is that individual factors are often well‐behaved and inexpensive to fit and sample in primary sample space, which leads to a practical, efficient sampling algorithm. Our sampling approach is unbiased, easy to implement, and compatible with multiple importance sampling. We show the results of applying our warps to projected solid angle sampling of spherical triangles, to sampling bilinear patch light sources, and to sampling glossy BSDFs and area light sources, with efficiency improvements of over 1.6× on real‐world scenes.

Smooth surfaces from rational bilinear patches

Smooth freeform skins from simple panels constitute a challenging topic arising in contemporary architecture. We contribute to this problem area by showing how to approximate a negatively curved surface by smoothly joined rational bilinear patches. The approximation problem is solved with help of a new computational approach to the hyperbolic nets of Huhnen-Venedey and Rörig and optimization algorithms based on it. We also discuss its limits which lie in the topology of the input surface. Finally, freeform deformations based on Darboux transformations are used to generate smooth surfaces from smoothly joined Darboux cyclide patches; in this way we eliminate the restriction to surfaces with negative Gaussian curvature.

Smooth surfaces from bilinear patches: Discrete affine minimal surfaces

Motivated by applications in freeform architecture, we study surfaces which are composed of smoothly joined bilinear patches. These surfaces turn out to be discrete versions of negatively curved affine minimal surfaces and share many properties with their classical smooth counterparts. We present computational design approaches and study special cases which should be interesting for the architectural application.

Programmable motion effects

Although animation is one of the most compelling aspects of computer graphics, the possibilities for depicting the movement that make dynamic scenes so exciting remain limited for both still images and animations. In our work, we experiment with motion depiction as a first-class entity within the rendering process. We extend the concept of a surface shader, which is evaluated on an infinitesimal portion of an object's surface at one instant in time, to that of a programmable motion effect, which is evaluated with global knowledge about all portions of an object's surface that pass in front of a pixel during an arbitrary long sequence of time. With this added information, our programmable motion effects can decide to color pixels long after (or long before) an object has passed in front of them. In order to compute the input required by the motion effects, we propose a 4D data structure that aggregates an object's movement into a single geometric representation by sampling an object's position at different time instances and connecting corresponding edges in two adjacent samples with a bilinear patch. We present example motion effects for various styles of speed lines, multiple stroboscopic images, temporal offsetting, and photorealistic and stylized blurring on both simple and production examples.

Ray Bilinear Patch Intersections

Abstract Ray tracing and other techniques employ algorithms which require the intersection between a three-dimensional parametric ray and an object to be computed. The object to intersect is typically a sphere, triangle, or polygon but many surface types are possible. In this work we consider intersections between rays and the simplest parametric surface, the bilinear patch. Unlike other surfaces, solving the ray-bilinear patch intersection with simple algebraic manipulations fails. We present a complete, efficient, robust, and graceful formulation to solve ray-bilinear patch intersections quickly. Source code is available online.

離散点で傾きが与えられたときの双一次パッチによる曲面形状決定法

This paper is concerned with a determination of a curved surface by use of a Bilinear patch when the gradients of the surface are given on discrete points. Equations expressing the gradients are derived with use of a Bilinear patch and solved by a least squares method. As the discrete points have no necessity to form a regular grid, this method can be applied to domains with arbitrary geometry. Applicability of this method is also confirmed in several cases.