Some applications of scalar and vector fields to geometric progressing of surfaces

Some applications of scalar and vector fields to geometric progressing of surfaces Jaime Puig-Pey, Akemi Gálvez, Andrés Iglesias, José Rodríguez, Pedro Corcuera, Flabio Gutiérrez Computers & Graphics 29 (2005) 723–729 Reporter: Yueqi Lu Thursday, December 17, 2005

Overview • About Author • Problem • Related Work & What to Do In this paper • Scalar and vector fields • Gradient curves on surfaces • point on a surface nearest to an external point • silhouette curve of a surface when observed from a given point • Final considerations

About Author • Jaime Puig-Pey, Akemi Gálvez, Andrés Iglesias, José Rodríguez, Pedro Corcuera • Department of Applied Mathematics and Computational Sciences, University of Cantabria, Spain • Flabio Gutiérrez • Department of Mathematics,National University of Piura,Perú

Problem • (1) point on a surface nearest to an external point;

Problem • (2) silhouette curve of a surface when observed from a given point

Related Work • General Geometric Problems: • BajajCL,Hoffmann CM,Lynch RE,Hopcroft JEH.[1988] • Krishnan S, Manocha D.[1997] : belonging, orthogonality,minimization of distances, etc.,by means of algebraic techniques

Related Work • Problems of this paper : • Grandine TA.[2000]:geometric problems are formulated as boundary problems in systems of algebraic-differential equations.

What to Do In this paper • initial value problems in systems of explicit first-order ordinary differential equations (ODEs) by means of geometric and differential arguments in all cases, • standard step-by-step numerical integration methods(Runge-Kutta methods and Matlab)

Scalar and vector fields • A scalar function: =(x1,x2,……,xn), X=(x1,x2,……,xn)  Rn • the vector field: Grad(), Grad()|P, the direction of maximum growing of when moving from that point P.

Scalar and vector fields • The gradient curves of scalar field: ………..(1) Taking an initial point Pi , the integration of (1) allows for the construction of a gradient line which passes throughPi

Scalar and vector fields • Implicit equation: The orientation of the trajectory along the gradient curve is selected by observing the sign of the  value at the initial point Pi.If it is positive, the orientation of -Grad() is taken, for maximum decrease of  from Pi, and if negative, one follows +Grad() for maximum increase.

Gradient curves on surfaces • P=(x,y,z) a point on surface S  R3, N the unit normal vector to the surface S atP, D(x,y,z)the vector value atPof vector fieldDdefined inR3, • T=[T1,T2,T3]=D-N(N•D) ……………(2)

Gradient curves on surfaces • The differential arc dC =[dx, dy, dz] of the gradient curve on Sassociated to vector field D, satisfies: dx/T1=dy/T2=dz/T3 …………….(3) • Getting a point Pi on the surface S as initial for a gradient curve, system (3) becomes an ODEs initial value problem, which defines that gradient curve passing Pi

Minimum distance between a surface and an external point • Q=[Q1,Q2,Q3] an extent point to the surface S, P=[x,y,z] an generic point on S • =(x-Q1)2+(y-Q2)2+(z-Q3)2 ………(4) • The condition of a relative minimum distance: the orientation of –Grad()

Minimum distance between a surface and an external point • Surface in implicit form: F(x,y,z)=0; D=[x - Q1, y - Q2, z - Q3]; ……….(5) N=[fx,fy,fz]/|| [fx,fy,fz] ||2 ; T=[T1,T2,T3]=D-N(N•D) ; ……… (2) dx/T1=dy/T2=dz/T3; ……….(3) ds2=dx2+dy2+dz2 ;

Minimum distance between a surface and an external point • Surface in implicit form: so we get: ……….(6)

Minimum distance between a surface and an external point • Surface in the parametric form: S(u,v)=[x(u,v),y(u,v),z(u,v)], (u,v)   R2 ; D=Grad(||S(u,v)-Q||22) =[2(S-Q).Su,2(S-Q).Sv]; …….(7) ds2=du2+dv2 ; With the –D vector in (7),we get: …….(8)

Minimum distance between a surface and an external point • Examples:

Silhouette curve on a surface • Surface in implicit form: • a surface S: F(x,y,z)=0; Q=[Q1,Q2,Q3] ,the view point; [x,y,z] ,a point of S on the silhouette curve; the following conditions must be satisfied:

Silhouette curve on a surface • Surface in implicit form:

Silhouette curve on a surface • Singular situations: W=0; • Initial point:

Silhouette curve on a surface • Surfaces in parametric form: • S(u,v)=[x(u,v),y(u,v),z(u,v)], (u,v)   R2 ; ………(14)

Silhouette curve on a surface • Surfaces in parametric form: • Initial point: border point: c(u, const)=0,or c(const,v)=0

Silhouette curve on a surface Example:

Final considerations • valid for smooth functions involved in the surface equations or surfaces consisting of several patches. • function ode45 of Matlab based on Runge–Kutta methods, absolute and relative error tolerances. • domain of definition of the x,y,z variables or for the u,v parameters. • numerical instabilities,such as with zero values for the partial derivatives

Thank you!

Some applications of scalar and vector fields to geometric progressing of surfaces