The fast identification of shadow regions due to area light sources is necessary for realistic rendering and for discontinuity meshing for global illumination. A new shadow-determination algorithm is presented that uses a data structure, called a backprojection, to represent the visible portion of a light source from any point in the scene. A complete discontinuity meshing algorithm is described for polyhedral scenes and area light sources, which includes an important class of light/geometry interactions that have not been implemented before. A fast incremental algorithm for computing backprojections is also described. The use of spatial subdivision, and heuristics based on computed statistics of typical scenes, results in efficient mesh and backprojection computation. Results of the implementation show that the use of the backprojection and discontinuity meshing permits accelerated high-quality rendering of shadows using both ray-casting and polygon-rendering with interpolants.

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A fast shadow algorithm for area light sources using backprojection

Gain valuable insight into better geometric approaches to high- degree surface patches by studying the simple free-form Steiner patch

Steiner Surface Patches

Steiner patches are triangular surface patches for which the Cartesian coordinates of points on the patch are defined parametrically by quadratic polynomial functions of two variables. It has recently been shown that it is possible to express a Steiner patch in an implicit equation which is a degree four polynomial in x,y,z. Furthermore, the parameters of a point known to be on the surface can be computed as rational polynomial functions of x,y,z. These findings lead to a straightforward algorithm for ray tracing Steiner patches in which the ray intersection equation is a degree four polynomial in the parameter of the ray. The algorithm presented represents a major simplification over existing techniques for ray tracing free-form surface patches.

Ray tracing of Steiner patches

Given a set $L$ of $n$ lines in ${\mathbb R}^3$, joints are points in ${\mathbb R}^3$ that are incident to at least three non-coplanar lines in $L$. We show that there are at most $O(n^{5/3})$ incidences between $L$ and the set of its joints.This result leads to related questions about incidences between $L$ and a set $P$ of $m$ points in ${\mathbb R}^3$. First, we associate with every point $p \in P$ the minimum number of planes it takes to cover all lines incident to $p$. Then the sum of these numbers is at most \[ O\big(m^{4/7}n^{5/7}+m+n\big).\] Second, if each line forms a fixed given non-zero angle with the $xy$-plane – we say the lines are equally inclined – then the number of (real) incidences is at most \[ O\big(\min\big\{m^{3/4}n^{1/2}\kappa(m),\ m^{4/7}n^{5/7}\big\} + m + n\big) , \] where $\kappa(m) \,{=}\, (\log m)^{O(\alpha^2(m))}$, and $\alpha(m)$ is the slowly growing inverse Ackermann function. These bounds are smaller than the tight Szemeredi–Trotter bound for point–line incidences in $\reals^2$, unless both bounds are linear. They are the first results of this type on incidences between points and $1$-dimensional objects in $\reals^3$. This research was stimulated by a question raised by G. Elekes.

Point–Line Incidences in Space

The tutorial presents some tools for free-form modeling with algebraic surfaces, that is, surfaces that can be defined using an implicit polynomial equation f(x, y, z)=0. Cubic algebraic surfaces (defined by an implicit equation of degree 3) are emphasized. While much of this material applies only to cubic surfaces, some applies to algebraic surfaces of any degree. This area of the tutorial introduces terminology, presents different methods for defining and modeling with cubic surfaces, and examines the power basis representation of algebraic surfaces. Methods of forcing an algebraic surface to interpolate a set of points or a space curve are also discussed. The parametric definition of cubic surfaces by imposing base points is treated, along with the classical result that a cubic surface can be defined as the intersection locus of three two-parameter families of planes. Computer-generated images of algebraic surfaces created using a polygonization algorithm and Movie. BYU software illustrate the concepts presented. >

Surfaces-techniques for cubic algebraic surfaces

Analytic geometry in three dimensions makes essential use of coordinate systems. To introduce a coordinate system, we consider triples (a,b, c) of real numbers, and we call the set of all such triples of real numbers the three-dimensional number space. We denote this space by R’. Each individual triple is a point in R 3 . The three elements in each number triple are called its coordinates. We now show how three-dimensional number space may be represented on a geometric or Euclidean three-dimensional space.

Analytic Geometry in Three Dimensions

We recall the elementary integration formula

$$\int_0^1 {{t^n}dt = } \left[ {\frac{1}{{n + 1}}{t^{n + 1}}} \right]_0^1 = \frac{1}{{n + 1}},$$

valid for any n >-1 Since need not be an integer, we employ the variable x and write 

$$\emptyset \left( x \right) = \int_0^1 {{t^x}dt} = \frac{1}{{x + 1}},x > - 1.$$

(1)

x>–1.

Murray H. Protter

Papers

Analytic Geometry in Three Dimensions

Differentiation under the Integral Sign. Improper Integrals. The Gamma Function