Part 4-Types and mathematical representations of Curves .pptx

Part #4
Types and Mathematical
Representations of Curves
1
Oct., 2022

Part #4-Outline
 Lecture 1:
 Introduction
 Types of Curves
 Representation of Curves
 Parametric Representation of
Curves
 Interpolation and
Approximation of Curve
 Properties for Curve Design
 Lecture 2 & 3:
 Synthesis (Free-Form) Curves
1) Hermite Curve
2) Bezier Curve
3) B-Spline Curves
4) Rational B-splines Curves
(including Non-uniform
rational B-splines –
NURBS)
 Lecture 4:
 Techniques for Surface
Modeling
1) Surface Patch
2) Coons and Bicubic
Patches
3) Bezier and B-Spline
Surfaces.

Objectives:
 Develop the various mathematical
representations of the curves and
surfaces used in geometric construction
 Understand the parametric
representation of curves and surfaces
and their relationship with computer
graphics.

Types, Representation &
Properties of Curves
 Lecture 1:
 Introduction
 Types of Curves
 Representation of Curves
 Parametric Representation of Curves
 Interpolation and Approximation of Curve
 Properties for Curve Design

Introduction
 This class presents the available types of entities of the modeling
technique and their related mathematical representations to enable good
understanding of how and when to use these entities in engineering
applications.
 Users usually have to decide on the type of modeling technique based on
the ease of using the technique during the construction phase and on the
expected utilization of the resulting database later in the design and
manufacturing processes.
 The most mathematically straight forward geometric entities are curves. A
curve is an integral part of any design and an engineer needs to draw one
or other type of curves or curved surfaces applicable to many engineering
components used in automotive, aerospace and hydrospace industries.
 Different types of shape constraints (e.g., continuity and/or curvature) are
imposed to accomplish specific shapes of the curve or curved surfaces.
 When a curve is two-dimensional, it lies entirely in a plane known as
planar curve. However, three-dimensional curve lies in space called space
curve.

Introduction
(a) Evenly Spaced data points (b) Non-evenly spaced data points

 Analytical Curve & Synthetic Curve are a two types of the curves used in
geometric modeling.
Analytical Curve: This types of curve can be represented by simple analytical
(mathematical) equations for which input is standard analytical mathematical
equation like point, line, arc, circle, ellipse, parabola & hyperbola. Conic curves or
conics are the curves formed by the intersection of a plane with a right circular cone
(parabola, hyperbola and sphere). These basic entities has a fixed form & cannot be
modified to achieve a shape that violates the mathematical equations. They can be
combined together with various end conditions to generate the overall curve design.
Types of curves
1) Analytical Curve:
A parabola is the curve created when a plane
intersects a right circular cone parallel to the
side (elements) of the cone.
An ellipse is the curve created when a plane cuts
all the elements ( sides ) of the cone but its not
perpendicular to the axis.

A hyperbola is the curve created when a plane parallel to the axis
and perpendicular to the base intersects a right circular cone.
This type of curve representation has the following advantages:
Types of curves
1) Analytical Curve:

Types of curves
2) Synthetic Curve:
Synthetic Curve: Unfortunately, it is not possible to represent all types of curves
required in engineering applications analytically; therefore, the method based on
the data points (synthetic curves) is very useful in designing the objects with
curved shapes. Products such as car bodies, ship hulls, airplane fuselage and
wings, propeller blades, shoe insoles, and bottles are a few examples that require
free-form, or synthetic curves and surfaces. It is computed by using geometric
input parameters like point, tangent. These parameters are processed to generate
curves such as Bezier , Hermite cubic, B-spline.

Types of curves
2) Synthetic Curve:

 Mathematically, curves can be described by two techniques:
1. Non-Parametric Equations can be explicit or implicit, and
2. Parametric Equations
1. Non-Parametric Equations:
Representation of curves

(1) Nonparametric equations must be solved simultaneously to
determine points on the curves, inconvenient process.
(2) If the slope of a curve at a point is vertical or near vertical, its
value becomes infinity or very large.
(3) However; the shapes of most engineering objects are intrinsically
independent of any coordinate system, the equations are
dependent on the choice of coordinate system.
(4) If the curve is to be displayed as a series of point or straight-line
segments, the computations involved could be extensive.
 Although non-parametric representation
of curve equations are used in some
cases, they are not in general suitable
for CAD. There are four problems with
describing curves using nonparametric
equations:

2. Parametric Curves:

Parametric Representation of Analytic Curves

1. Lines:

2. Circles:

 The analytical form of planar curves is not suitable for designing the complex three-
dimensional curves and surfaces used for designing the complex shaped objects.
 The need for synthetic curves in design arises on two occasions; when a curve is represented
by a collection of measured data points and when an existing curve must change to meet
new design requirements.
 The designer prefers the synthetic curve, which passes through the set of data points,
because designer has full control on its shape as per the new design requirements.
 A spline curve is defined by giving a set of coordinate positions, call control points, which
indicate the general shape of the curve. These control points are then fitted with piecewise
continuous parametric polynomial functions.
 Mathematically, curve fitting (interpolation) and curve fairing (approximation) techniques
are used for generating the curves & surfaces in CAD:
1. Interpolation Techniques: If the problem of curve design is a problem of data fitting,
the classic interpolation solutions are used. When polynomial sections are fitted so that
the curve passes through all the points, or through each control point, the resulting curve
is said to interpolate the set of control points.
2. Approximation Techniques: On the other hand, if the problem is dealing with free-form
design with smooth shapes, approximation methods are used. when the polynomials are
fitted to the general control point path without necessarily passing through any control
point, the resulting curve is said to approximate the set of control points.
Interpolation and Approximation of Curve:

Interpolation and Approximation of data points (a) curve fitting (b) curve fairing

Properties for Curve Design
(a) Zero Order Continuity (b) First Order continuity (c) Second Order Continuity

 Lecture 2 & 3:
 Synthesis (Free-Form) Curves
1) Hermite Curve
2) Bezier Curve
3) B-Spline Curves
4) Rational B-splines Curves (including Non-
uniform rational B-splines – NURBS)
Synthesis (Free-Form)
Curves

Synthesis (Free-Form) Curves:
Spline Curves

Classifications of Spline Curves

1) Hermite Curves
Shape Control

2) Bezier Curves
 In fact, two outside points control the shape of Bézier curves

2) Bezier Curves
Effect of position of control points on the shape of cubic Bézier curves

2) Bezier Curves
Blending Functions Formulation

2) Bezier Curves
Blending Functions Formulation
(a) Bézier/Bernstein blending functions for three
control points
(b) Bézier/Bernstein blending functions for four
control points
(c) Bézier/Bernstein blending functions for five control points

2) Bezier Curves
 The properties of Bézier curves depend upon the properties of Bernstein polynomial.
Properties of Bézier Curves

2) Bezier Curves
 The properties of Bézier curves are:

2) Bezier Curves

2) Bezier Curves
This represents that the sequence of control points defining the curve can be
changes without modify of the curve shape.

2) Bezier Curves
Composite Bézier Curves

2) Bezier Curves
Drawbacks of Composite Bézier Curves:

Essential Requirements for Synthetic Curves

3) B-Spline Curves
Types of B-spline Curves:

Techniques for Surface Modeling
 Lecture 4:
 Techniques for Surface Modeling
1) Surface Patch
2) Coons and Bicubic Patches
3) Bezier and B-Spline Surfaces.

Part 4-Types and mathematical representations of Curves .pptx

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