Generalized Distance Functions ERGUN AKLEMAN Visualization Sciences College of Architecture Texas A&M University College Station, Texas 77843-3137 email: ergun@viz.tamu.edu phone: +(409) 845-6599 fax: +(409) 845-4491 J IANER CHEN Department of Computer Science College of Engineering Texas A&M University College Station, TX 77843-3112 email: chen@cs.tamu.edu phone: +(409) 845-4259 fax: +(409) 847-8578 Abstract In this paper, we obtain a generalized version of the well- known distance function family L p norm. We prove that the new functions satisfy distance function properties. By using these functions, convex symmetric shapes can be described as loci, the set of points which are in equal distance from a given point. We also show that these symmetric convex shapes can be easily parameterized. We also show these distance functions satisfy a Lipschitz type Condition. We provide a fast ray marching algorithm for rendering shapes described by these distance functions. These distance func- tions can be used as building blocks for some implicit mod- eling tools such as soft objects, constructive soft geometry, freps or ray-quadrics. 1 Introduction and Significance The concept of distance function has been developed to provide a formal description for measuring distance be- tween two points in a vector space. Any function which satisfies the following logical conditions for distance can be used for measuring distance in a vector space. Definition 1.1 Let V be a vector space and ℜ * be the set of all non-negative numbers, a function f : V−→ℜ * is a distance function, if it satisfies the following distance con- ditions: (1) f (v)=0 if and only if v =0; (2) f (v)= f (−v); (3) f (v 1 + v 2 ) ≤ f (v 1 )+ f (v 2 ) for any v 1 ,v 2 ∈V . There exist various distance functions defined over different vector spaces such as L p norm, Hausdorf- Besicowitch distance, and Hamming distance. L p norm is a particularly interesting distance function for modeling shapes. In 3-dimensional vector space, L p is given as f (x, y, z)=(|x| p + |y| p + |z| p ) 1/p . L p norm is based on the operator ( ∑ r i=1 () p ) 1/p which we call Minkowski operators. These operators satisfy well- known Minkowski’s inequality. Interested readers can find a proof for this inequality in many classical mathematics textbooks (e.g., [15], page 55). Proposition 1.1 (Minkowski) If a i ≥ 0 and b i ≥ 0, for i =1,...,r, and p ≥ 1, then  r  i=1 (a i + b i ) p  1/p ≤  r  i=1 a p i  1/p +  r  i=1 b p i  1/p The Minkowski operators have been used in implicit shape modeling in quite a long time. Ricci [20] extended these operators by including negative p values and ob- tained exact and approximate set operations which we call Ricci operators. Barr [3], independently, has developed su- perquadrics by using Minkowski operators. Hanson [11] also used Minkowski operators to generalize superquadrics to Hyperquadrics. Akleman [1] developed ray-linears, a function family that provide fast computation and closed under Ricci operators. Wyvill [24] has developed Con- structive Soft Geometry also by using Ricci’s operators. Even Rvachev’s exact set operations [17] are related to Minkowski operators. Distance functions are also useful to generate implicit field functions from various types of shape information [13, 8, 9, 10]. In this paper, we obtain new distance functions also using Minkowski operations. Distance functions are interesting for shape modeling for many reasons.