# Parametric representation and polygonal decomposition of curved surfaces

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Jun 01,  · In this section we will take a look at the basics of representing a surface with parametric equations. We will also see how the parameterization of a surface can be used to find a normal vector for the surface (which will be very useful in a couple of sections) and how the parameterization can be used to find the surface area of a surface.

Parametric Representations of Surfaces Part 1: Parameterizing Surfaces.

Surfaces in three dimensional space can be described in many ways -- for example, Use the cylindrical coordinates u = and v = z to construct a parametric representation of a circular cylinder of radius 2 and height 3.

Plot your parametric surface in your worksheet. Abstract. This paper describes an algorithm for the polygonal subdivision of parametric surfaces. Our existing method has problems displaying trimmed surfaces with complex boundaries or surfaces with tiny holes, because parametric surfaces are divided into fine polygons of uniform jacksonmealsmatter.com by: 1.

### Description Parametric representation and polygonal decomposition of curved surfaces EPUB

A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters →: →. Parametric representation is a very general way to specify a surface, as well as implicit jacksonmealsmatter.comes that occur in two of the Parametric representation and polygonal decomposition of curved surfaces book theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form.

MAE Computer-Aided Design and Drafting Parametric Representation of Curves and Surfaces How does the computer compute curves and surfaces. Parametric Surfaces. Substitution Recall that a curve in space is given by parametric equations as a function of single parameter t x= x(t) y= y(t) z= z(t): A curve is a one-dimensional object in space so its parametrization is a function of one variable.

Analogously, a surface is a two-dimensional object in space and, as such can be described. Parametric Representation A parametric representation of a function expresses the functional relationship between several variables by means of auxiliary variable parameters.

In the case of two variables x and y, the expression F(x, y) = 0 can be geometrically interpreted as the equation of a plane curve. Any variable t that determines the position of a.

Parametric representation of a surface pencil with an asymptotic curve. Suppose we are given a 3-dimensional parametric curve r (s), L 1 ≤ s ≤ L 2, in which s is the arc length (regular and ‖ r ′ (s) ‖ = 1, L 1 ≤ s ≤ L 2) and r (s) has second jacksonmealsmatter.com by: Parametric Representations of Surfaces Part 2: Local Change-in-Area Factors.

It is often useful to convert from one set of parameters to another. This conversion is called a change of coordinates and can be expressed as a pair of functions from one set of parameters, or coordinates, to the other set. Parametric curves CS Computer Graphics 6 Convex hull property Convex set: A convex set is a collection of points in which the line connecting any pair of points in the set lies entirely within the set.

Convex Hull: Given a collection of points, the convex hull is. Analytic representation of surfaces Similar to the curve case there are mainly three ways to represent surfaces, namely parametric, implicit and explicit methods.

In parametric representation the coordinates of a point of the surface patch are expressed as functions of the parameters and in a closed rectangle. A PARAMETRIC REPRESENTATION OF RULED SURFACES 3 2. Ruled surfaces A ruled surface can be generated by the motion of a line in space, similar to the way a curve can be generated by the motion of a point.

A 3D surface is called ruled if through each of its points passes at least one line that lies entirely on that surface. Offsets of Parametric Curves and Surfaces. • Curved plate (shell) representation in This paper describes a method to extract the generic features of free-form parametric surfaces for.

Jun 04,  · Here is a set of practice problems to accompany the Parametric Surfaces section of the Surface Integrals chapter of the notes for Paul Dawkins Calculus IiI course at Lamar University.

13 Piecewise Linear Approximation of Surface Triangle Mesh Polygons are easier to process and display than curved surfaces. 14 Polygon Meshes Set of edge-connected planar polygons (usually triangles or quads) Faces share vertices and edges To avoid repeating vertices, store each vertex once Each face stored as set of indices into the vertex list.

Surfaces and solids CS Lecture 17 1. – normally use parametric representation – line—just a point and a vector (like ray in ray tracer) machinery for several ways of building curved surfaces • Building surfaces from 2D curves – extrusions and surfaces of revolution.

This paper proposes a simple parametric system to generate an almost complete set of ruled surfaces that may be used to describe building geometry. The major classes of regular, named ruled surfaces Cited by: 4. Mar 07,  · For the Love of Physics - Walter Lewin - May 16, - Duration: Lectures by Walter Lewin.

They will make you ♥ Physics. Recommended for you. 76 CHAPTER IMPLICIT AND PARAMETRIC SURFACES Implicit representations of surfaces An implicit representation takes the form F(x) = 0 (for example x2 +y2 +z2 r2 = 0), where x is a point on the surface implicitly described by the function F.

In other words, point x is on the surface if and only if the relationship F(x) = 0. Jan 03,  · In this video we explore Parametric Surfaces and their Areas. We will begin with a brief review of how to parameterize a curve, which is when we take a value for time along an interval and plug it into our vector to find where we are on curve.

Petruševski et al.: Parametric curves and surfaces MATHEMATICA demonstrations as a tool in exploration of architectural form spatium 69 Demonstration “Four Space Curves”, shown in Figure 3, describes parametric equations of circle, Archimedes’s spiral, helix and conical spiral.

78 CHAPTER IMPLICIT AND PARAMETRIC SURFACES Implicit representations of surfaces An implicit representation takes the form F(x) = 0 (for example x2 +y2 +z2 r2 = 0), where x is a point on the surface implicitly described by the function F.

In other words, point x is on the surface if and only if the relationship F(x) = 0. Polygon Meshes and Implicit Surfaces Polygon Meshes Parametric Surfaces Implicit Surfaces Constructive Solid Geometry Polygon Meshes Parametric Surfaces Implicit Surfaces Constructive Solid Geometry 10/01/ 2 Modeling Complex Shapes • We want to build models of very complicated objects.

Apr 01,  · Homework Statement I am trying to find parametric representation of the right surface of a sphere which was cut along the line y=5. x^2 + y^2 + z^2 = 36 Homework Equations The Attempt at a Solution x^2 + y^2 + z^2 = 36 This is an equation of a sphere with radius given by: r^2 = 36 r=6.

Parametric surfaces. This is the currently selected item. Vector fields, introduction. Fluid flow and vector fields. 3d vector fields, introduction.

3d vector field example. Next lesson. Transformations. Video transcript - So I have here a very complicated function. It's got a two-dimensional input, two different coordinates to its input, and. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian jacksonmealsmatter.comes have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within.

All the parameterizations we've done so far have been parameterizing a curve using one parameter.

### Details Parametric representation and polygonal decomposition of curved surfaces EPUB

What we're going to start doing this video is parameterizing a surface in. Parametric representation of a surface pencil with a line of curvature. First, we recall a known theorem, which characterizes lines of curvature on a surface.

Theorem 1. A necessary and sufficient condition that a curve on a surface be a line of curvature is that the surface normals along Cited by: We have dealt extensively with vector equations for curves, ${\bf r}(t)=\langle x(t),y(t),z(t)\rangle$.

A similar technique can be used to represent surfaces in a way that is more general than the equations for surfaces we have used so far. Survey of Polygonal Surface Simplification Algorithms. generated by subdivision of curved parametric surfaces.

A Theory of Multiresolution Signal Decomposition: The Wavelet Representation. Trimmed NURBS surfaces Uniform B-spline surfaces are a special case of NURBS surfaces.

Sometimes, we want to have control over which parts of a NURBS surface get drawn. For example: We can do this by trimming the u-v domain. Define a closed curve in the u-v domain (a trim curve) Do not draw the surface points inside of this curve.Parametric Curves & Surfaces Based on lecture notes by Greg Humphreys, UoV MSc Computer Games and Entertainment Maths & Graphics II Lecturers: FFL (in collaboration with Gareth Edwards) Curved Surfaces –More concise representation than polygonal mesh.

CONTEXT.• Each one of them divides the representation space into infinite portions, one filled with material and the other empty •Surfaces can be considered half-space boundaries and half spaces can be considered directed surfaces •An object is defined by the volume space contained within.