Ship Computer Aided Design

Slides:



Advertisements
Similar presentations
Boyce/DiPrima 9th ed, Ch 2.4: Differences Between Linear and Nonlinear Equations Elementary Differential Equations and Boundary Value Problems, 9th edition,
Advertisements

Arc-length computation and arc-length parameterization
Splines I – Curves and Properties
© University of Wisconsin, CS559 Spring 2004
Differential geometry I
Parametric Equations Local Coordinate Systems Curvature Splines
November 12, 2013Computer Vision Lecture 12: Texture 1Signature Another popular method of representing shape is called the signature. In order to compute.
Lecture 10 Curves and Surfaces I
Chapter 13 – Vector Functions
Geometric Modeling Notes on Curve and Surface Continuity Parts of Mortenson, Farin, Angel, Hill and others.
Shanghai Jiao Tong University 1 UG: Free Form Feature ME 250: Design & Manufacturing I School of Mechanical Engineering.
Basis Expansion and Regularization Presenter: Hongliang Fei Brian Quanz Brian Quanz Date: July 03, 2008.
Dr. S.M. Malaek Assistant: M. Younesi
EARS1160 – Numerical Methods notes by G. Houseman
Slide 127 October 1999CS Computer Graphics (Top Changwatchai) Review of Spline Concepts Sections 10-6 to in Hearn & Baker Splines can be 2D.
Rational Bezier Curves
Offset of curves. Alina Shaikhet (CS, Technion)
Modelling. Outline  Modelling methods  Editing models – adding detail  Polygonal models  Representing curves  Patched surfaces.
CS485/685 Computer Vision Prof. George Bebis
Implicit Surfaces Tom Ouyang January 29, Outline Properties of Implicit Surfaces Polygonization Ways of generating implicit surfaces Applications.
Curves Locus of a point moving with one degree of freedom
Geometric Modeling Surfaces Mortenson Chapter 6 and Angel Chapter 9.
Curves Mortenson Chapter 2-5 and Angel Chapter 9
Drawing Parametric Curves Jean-Paul Mueller. Curves - The parametric form of a curve expresses the value of each spatial variable for points on the curve.
Chapter 16 – Vector Calculus
1 Representing Curves and Surfaces. 2 Introduction We need smooth curves and surfaces in many applications: –model real world objects –computer-aided.
Parts of Mortenson Chapter 6-9,
Curves and Surfaces (cont’) Amy Zhang. Conversion between Representations  Example: Convert a curve from a cubic B-spline curve to the Bézier form:
Curve Modeling Bézier Curves
Computer Graphics: Programming, Problem Solving, and Visual Communication Steve Cunningham California State University Stanislaus and Grinnell College.
1 Geometry. 2 Objectives Introduce the elements of geometry ­Scalars ­Vectors ­Points Develop mathematical operations among them in a coordinate-free.
Conics, Parametric Equations, and Polar Coordinates
A D V A N C E D C O M P U T E R G R A P H I C S CMSC 635 January 15, 2013 Spline curves 1/23 Curves and Surfaces.
October 14, 2014Computer Vision Lecture 11: Image Segmentation I 1Contours How should we represent contours? A good contour representation should meet.
V. Space Curves Types of curves Explicit Implicit Parametric.
Lecture 5: Curves And Surfaces, and Geometric Modeling
Curves.
Ken YoussefiME Dept. 1 Surface Modeling in SW Open sketch (spline) Starting with a sketch, you can create a surface using Extrude, Revolve, Loft, Boundary.
PREPARED BY: SAMERA BINTI SAMSUDDIN SAH SEM /2012 (NOV 2011)
Quadratic Surfaces. SPLINE REPRESENTATIONS a spline is a flexible strip used to produce a smooth curve through a designated set of points. We.
Geometric Modeling. Volumetric o Collection device obtains regular grid of measurement values Examples: CT, MRI, PET, Ultrasound o Values are interpreted/visualized.
Course 13 Curves and Surfaces. Course 13 Curves and Surface Surface Representation Representation Interpolation Approximation Surface Segmentation.
Vector Calculus CHAPTER 9.10~9.17. Ch9.10~9.17_2 Contents  9.10 Double Integrals 9.10 Double Integrals  9.11 Double Integrals in Polar Coordinates 9.11.
Parametric Surfaces Define points on the surface in terms of two parameters Simplest case: bilinear interpolation s t s x(s,t)x(s,t) P 0,0 P 1,0 P 1,1.
Copyright © Cengage Learning. All rights reserved. 16 Vector Calculus.
Ship Computer Aided Design MR 422. Geometry of Curves 1.Introduction 2.Mathematical Curve Definitions 3.Analytic Properties of Curves 4.Fairness of Curves.
Basic Theory (for curve 02). 1.3 Parametric Curves  The main aim of computer graphics is to display an arbitrary surface so that it looks real.  The.
04/18/02(c) 2002 University of Wisconsin Last Time Hermite Curves Bezier Curves.
Interactive Graphics Lecture 10: Slide 1 Interactive Computer Graphics Lecture 10 Introduction to Surface Construction.
11/6/ :55 Graphics II Introduction to Parametric Curves and Surfaces Session 2.
Greg Humphreys CS445: Intro Graphics University of Virginia, Fall 2003 Parametric Curves & Surfaces Greg Humphreys University of Virginia CS 445, Spring.
Splines I – Curves and Properties based on: Michael Gleicher Curves, chapter 15 in Fundamentals of Computer Graphics, 3 rd ed. (Shirley & Marschner) Slides.
CAP 4703 Computer Graphic Methods Prof. Roy Levow Chapter 10.
CS 450: Computer Graphics PARAMETRIC SPLINES AND SURFACES
Splines Sang Il Park Sejong University. Particle Motion A curve in 3-dimensional space World coordinates.
Ship Computer Aided Design
Basic Theory (for curve 01). 1.1 Points and Vectors  Real life methods for constructing curves and surfaces often start with points and vectors, which.
CHAPTER TWO : Geometric Tolerances
Computer Graphics I, Fall 2010 Geometry.
Ken YoussefiME Dept. 1 Surface Modeling in SW Open sketch (spline) Starting with a sketch, you can create a surface using Extrude, Revolve, Loft, Boundary.
Rick Parent - CIS681 Interpolation of a time-space curve Interpolation (of a curve in space) Speed control along curve Reparameterizing according to arc.
Object Modeling: Curves and Surfaces CEng 477 Introduction to Computer Graphics.
CHAPTER 9.10~9.17 Vector Calculus.
Parametric equations Parametric equation: x and y expressed in terms of a parameter t, for example, A curve can be described by parametric equations x=x(t),
Curve & Surface.
Copyright © Cengage Learning. All rights reserved.
Basic theory of curve and surface
Three-Dimensional Object Representation
Spline representation. ❖ A spline is a flexible strip used to produce a smooth curve through a designated set of points. ❖ Mathematically describe such.
Presentation transcript:

Ship Computer Aided Design MR 422

Geometry of Surfaces Introduction Mathematical Surface Definitions: Parametric vs. Explicit vs. Implicit. Analytic Properties of Parametric Surfaces Surface Curvature Continuity Between Surfaces Fairness of Surfaces Spline Surfaces

1. Introduction A surface is a 2-D continuous point set embedded in a 2D or 3-D space. Surfaces may be used as: Explicit design elements, such as the hull or weather deck surfaces Construction elements, such as a horizontal rectangular surface locating an interior deck Boundaries for solids.

2. Mathematical Surface Definitions Defining surfaces mathematically ways are: Implicit surface definition Explicit surface definition Parametric surface definition

f(x, y, z) = (x _ a)2 + (y _ b)2 + (z _ c)2 - r2 = 0 . Implicit surface definition: A surface is defined in 3-D as the set of points that satisfy an implicit equation in the three coordinates: f(x, y, z) = 0 Homogeneous, free of the coordinate singularities but Lacks any natural surface coordinate system Example: a complete spherical surface is defined as the set of points at a given distance r from a given center point {a, b, c}: f(x, y, z) = (x _ a)2 + (y _ b)2 + (z _ c)2 - r2 = 0 .

Explicit surface definition: In 3-D, one coordinate is expressed as an explicit function of the other two, for example: z = f(x, y) have seen some use in ship form definitions, but usually have problems with some shapes which restrict the range of shapes that can be accommodated without encountering mathematical singularities.   Example of explicit definition of ship hull forms is the series of algebraic shapes, commonly called the “Wigley parabolic hull”.

The waterlines (z= constant) and underwater sections (x= constant) are families of parabolas. These simple explicit surface equations allowed the computation of Michell’s integral analytically, allowing an early comparison of this influential theory with towing-tank results.

x = f(u, v), y = g(u, v), z = h(u, v) Parametric surface definition: In either 2-D or 3-D, each coordinate is expressed as an explicit function of two common dimensionless parameters: x = f(u, v), y = g(u, v), z = h(u, v) The parametric surface can be described as a locus in three different ways: The locus of a moving point {x, y, z} as the parameters u, v vary continuously over a specified domain such as [0, 1] X [0, 1], or The locus of a moving parametric curve (parameter u or v) as the other parameter (v or u) varies continuously over a domain such as [0, 1]. Subdivision surfaces. Parametric surface definitions avoid the limitations of implicit and explicit definitions and are widely employed in 3-D CAD systems today

3. Analytic Properties of Parametric Surfaces Definitions: x(u, v) : a parametric surface A bold face letter signifies a vector of three components. Assume the range of each parameter u, v is [0, 1] The bounded surface patch corresponds to the nominal parameter range. The parameter space of the surface: is the 2-D space of u and v. The 3-D surface is a mapping of the parameter-space points into three-space points, moderated by the surface equations x(u, v).

The unit normal vector varies with u and v unless the surface is flat. The tangent plane is the plane passing through a surface point, normal to the unit normal vector at that point. The direction of the unit normal on, for example, one of the wetted surfaces of a ship may be inward (into the hull interior) or outward (into the water), depending on the orientation chosen for the parameters u, v. For many purposes the normal orientation will not matter; however, for other purposes it is of critical importance.

For hydrostatic or hydrodynamic analysis, it is usually necessary to create panels having a consistent orientation of corner points, e.g., counterclockwise when viewed from the water; this may well require that the surface normal have a prescribed orientation. The bevel angles: the angles of the unit normal with respect to the coordinate planes, are sometimes required during construction. The angle between n and the unit vector in the x direction is most often used

4. Surface Curvature At a point P on a surface S, where S is sufficiently smooth (i.e., a unique normal line N and tangent plane T exist), several measures of surface curvature can be defined. Each plane that cuts the surface S in a plane curve C, known as a normal section. The curvature of C at P is called a normal curvature (kn) of S (dimensions 1/length) at this location.

Normal curvature depends on α Normal curvature depends on α. As α varies, kn varies sinusoidally with respect to α, and in general goes through maximum and minimum values k1,k2 (the principal curvatures). The directions of the two principal curvatures are orthogonal, and are called the principal directions.  The product k1 k2 of the two principal curvatures is called Gaussian curvature K (dimensions 1/length2). The average (k1 +k2)/2 of the two principal curvatures is called mean curvature H (dimensions 1/length).

5. Continuity Between Surfaces Levels of geometric continuity are defined as follows: G0: Surfaces that join with an angle or knuckle (different normal directions) at the junction. G1: Surfaces that join with the same normal direction at the junction. G2: Surfaces that join with the same normal direction and the same normal curvatures in any direction that crosses the junction.

G0 continuity easy to achieve, used in “industrial” contexts when a sharp corner does not interfere with function, for example, the longitudinal chines of a typical metal workboat.   G1 continuity widely used in industrial design when rounded corners and fillets are functionally required, for example, a rounding between two perpendicular planes achieved by welding in a quarter-section of cylindrical pipe. G2 continuity still more difficult to attain, is required for the highest levels of visual design, as in automobile and yacht exteriors.

6. Fairness of Surfaces Fairness is best described as the absence of certain kinds of features that would be considered unfair: surface slope discontinuities (creases, knuckles) local regions of high curvature (e.g., bumps and dimples) flat spots (local low curvature) abrupt change of curvature (adjoining regions with less than G2 continuity) unnecessary inflection points

Fairness in the longitudinal direction receives more emphasis than in the transverse direction. Thus, for example, longitudinal chines are tolerated for ease of construction, but transverse chines are very much avoided (except as steps in a high speed planning hull, where the flow deliberately separates from the surface). To reveal unfairness of physical surfaces:  Reflection lines: Viewing the reflections that occur at low angles (assuming a polished, reflective surface) of a regular grid, can be computed and presented in computer displays to simulate this process using the visualization technology known as ray tracing.  Highlight lines: contours of equal “slope” s on the surface.

7. Spline Surfaces A spline surface is typically divided along certain parameter lines (its knotlines) into sub-surfaces or spans, each of which is a parametric polynomial (or rational polynomial) surface in u and v. Within each span, the surface is analytic (continuous derivatives of all orders) At the knotlines, the spans join with levels of continuity depending on the spline degree. Cubic spline surfaces have C2 continuity across their knotlines, which is generally considered adequate continuity for all practical visual and hydrodynamic purposes. Splines of lower order than cubic (i.e., linear and quadratic) are simpler to apply and provide adequate continuity (C0 and C1, respectively) for many less demanding applications.

Interpolating Spline Lofted Surface Interpolating spline curves passes through an arbitrary set of data points. A lofted surface interpolates an arbitrary set of parent curves, known as master curves or control curves.