Examining of Tetragonal Surface Patches with Their Geometric Properties and Relationships

In this paper, the geometric properties and relationships of the rectangular are applied to a tetragonal surface patch. Then, O. Bonnet integral formula is generalized for the tetragonal surface patchs which are bounded with a curvatural polygon.


INTRODUCTION
This article presents the geometric properties and relationships of rectangulars in a plane to apply tetragonal patches on surface. Therefore, in the second section; using some formulas of the surface geometry, [3], [6], [8], and especially J. Liouville and O. Bonnet integral formulas, [1], [2], [4], [5], are reminded. In the third section, the method of examining of rectangulars in the plane, [7], is generalized to a tetragonal surface patch which is divided to mxn sub-patches. Then, using formulas that are given in the second section, O. Bonnet integral formula is generalized for the area of the tetragonal surface patch.

SURFACE GEOMETRY Definition 2.1
A subset M ⊂ R 3 is a regular surface if, for each M p ∈ , there exists a neighborhood V in R 3 and a map r : U → V ∩ M of an open set U ⊂ R 3 onto V ∩ M ⊂ R 3 such that 1-r is differentiable. This means that if we write r(u, v)=(x (u, v), y (u, v), z(u, v)), (u, v) ∈U the function x(u, v), y (u, v), z (u, v) have continuous partial derivatives of all orders in U.
2-r is homeomorphism. Since r is continuous by condition 1, this means that r has an inverse 1 3-(The regularity condition) For each q∈U, the differential : The mapping r is called a parameterization or a system of (local) coordinates in (a neighborhood of ) p. The neighborhood V ∩ M of p in M is called a coordinate neighborhood.
To give condition 3 a more familiar form, let us compute the matrix of the linear map q dr in the canonical bases 1 e = (1,0), 2 e = (0,1) of R 2 with coordinates (u, v) and Explicitly, for each point ( By the definition of differential [3], [8].

Definition 2.2 (The First and Second Fundamental Form of the Surface)
Let M be a surface is given by r r = r r (u, v), then u and v are called the curvilinear coordinates of this surface. The vector equation of the curve on surface and its differential are The first fundamental form, as ds is differential of its length, is The unit normal vector of the surface and its differential are then the second fundamental form is [2], [3], [5], [6].

Definition 2.3
If tangent vector α at a point P of a (α ) curve on a M ⊂ E 3 surface is parallel to one of the principal directions of M surface, then (α ) is called a principal curve or curvature line. When Frenet and Darboux frames on curvature lines are taken as ( b n t r r r , , ), ) respectively, we have where s is the arc length of the curve, θ is the angle between principal normal and surface normal, k n is normal curvature and τ is the torsion, [3], [6].

Theorem 2.2
Let us assume that ϕ is the angle between the unit tangent vector t r that passes through a point P of a (α) curve on a M surface and the first of 1 t r and 2 t r unit tangent vectors of the parameter curves at this point, we have where k n1 and k n2 are principal curvatures and the normal curvatures corresponding to t r is k n and geodesic torsion is τ g , [1].

Theorem 2.3
Let us assume that the trihedrons of u = const and v = const parameter curves, which are perpendicular to each other, passes through a point P on a surface M and any curve of the surface passes through P are ( where k g1 , k g2 and τ g1 , τ g2 are geodesic curvatures and torsions that belongs to parameter curves, respectively, [1].

Theorem 2.4 (Formula of J. Liouville)
Let 1 t r and 2 t r be the tangent directions of v = const, u = const parameter curves which are perpendicular to each other at a point P on the surface r r = r r (u, v) and the geodesic curvature of these directions are k g1 and k g2 , respectively. Let t r be a tangent direction of any curve through point P on the surface. Then, we get the relation of the geodesic curvature of t r direction ϕ ϕ ϕ sin cos where ϕ is the angle between t r and 1 t r , [3], [4].

Theorem 2.5 (O.Bonnet's Integral Formula)
Let A be a region with single dependence of M surface that is given by r r = r r (u, v) occurs regular points. Let (α) be a curve, which is continuous and without multiple point, where is limit to this surface area, and k g be geodesic curvature of the surface through this curve. Also, let K be Gauss curvature which is belong to points on this surface region and if the arc element of the curve is ds and the surface element is dσ which is belong to region that occurs the given surface, then we have The integral is taken on positive direction on the curve (α), [2], [4].

Definition of Tetragonal Surface Patches
Describe a tetragonal surface patch by giving the coordinate curve of one of its vertices and the lengths of its two principal curves. Consider only tetragonal surface patches whose sides are parallel to the coordinate axes.
Denote a tetragonal surface patch symbolically with the boldface capital letters TSP. Again, indicate a specific tetragonal by adding an alphabetic or numeric subscript : TSP 1 , for example. Thus, to specify a tetragonal surface patch, we write  If the tetragonal surface patch divided to mxn sub-patches, then the length of the sides are m and n. So, to specify a tetragonal surface patch, we write ). In counterclockwise order from the minimum vertex, they are p 1 ) , ( Again use a lowercase boldface p to denote a point and a subscript to indicate a specific point, as in Figure 1. Note that p min = p 0 and p max = p 2 .

Generalization of The O. Bonnet Integral Formula For Tetragonal Surface Patches
If (α) curve is a curvatural polygon which is formed of some regular arc pieces coming together as shown in Figure 2, the function ϕ at edge points of this curvatural polygon makes a stepping as much as the external angel at the edge point, for instance the angle ϕ is passing from

Figure 2
However, if β i = π−α i , we get Then, the integral formula of O.Bonnet (9) is formed by where n is the number of curvatural polygon's edges.
In E 3 , we find If we take the polygon as a linear polygon, then k 1 →0 and we get To find the length of the domain of the tetragonal surface patch, we firstly find

The Relationships And Algorithms of The Tetragonal Surface Patch With a Point And The Other Tetragonal Surface Patches
The geometric center of a patch is at the point Many computer graphics problems require that you determine if a given point is inside, outside, or on the boundary of a closed shape. Given a point p = (u, v), test its coordinates against those of p min and p max . Figure 3, if the point is inside the patch, then both of the following conditions must be satisfied : u min < u < u max and v min < v < v max . If the point is outside the patch then at least one of the following conditions is true : u < u min or u > u max , v < v min or v > v max . Let p min,1 (u min,1 , v min,1 ) and p max,1 (u max,1 , v max,1 ) are the minimum and maximum points of the patch TSP 1 , respectively. Also, Similarly, let p min,2 (u min,2 , v min,2 ) and p max,2 (u max,2 , v max,2 ) are the minimum and maximum points of the patch TSP 2 , respectively. Also,