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CURVES

CURVES. Curves are used to draw wireframe models, surface models and solid models. (All these models are made up of points, and curves) Curves can be described in two ways Arrays of co-ordinate data Analytic equations. Disadvantages of the Co-ordinate System .

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CURVES

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  1. CURVES Curves are used to draw wireframe models, surface models and solid models. (All these models are made up of points, and curves) Curves can be described in two ways • Arrays of co-ordinate data • Analytic equations

  2. Disadvantages of the Co-ordinate System. • Storage Required : Excessively large • Computation: To transform data from one form to another is cumbersome • Exact shape: Not known. Hence calculations of intersection of curves, physical properties of objects like volume calculations, etc., are difficult. • Redesign Point of View: Very difficult.

  3. Classification of Curves. • Analytical Curve: Represented by a simple mathematical eqn. Eg: circle or ellipse - Has a fixed form and cannot be modified to achieve a shape that violates mathematical principles - Provides designers with info on Effective set of data points, Curve behaviour, control, continuity and curvature .

  4. Classification of Curves. (Contd.) 2. Interpolated Curves: Drawn by interpolating the given data points. - Has a fixed form, dictated by the data points. - Limited flexibility in shape creation, also dictated by data points. 3. Approximated Curves: Provide most flexibility for drawing complex shapes. Eg: Model of a curved automobile fender can be made with approximated curves and surfaces

  5. In general, a fixed form curve will generate a fixed form surface only. i.e., the generated surface is of the same type as the generating curve. Computer Representation: To facilitate computer language algorithm, curves are represented by Parametric or Non-Parametric equations.

  6. Non - Parametric Equations. (Explicit) A non-parametric equation has the form, y = c1 + c2 x + c3 x2 + c4 x3 • This is an Explicit non-parametric equation. In this equation, there is a unique single value of the dependent variable for each value of the independent variable. E.g.: For a circle: x2 + y2 = r2

  7. Non- Parametric Equation.( Implicit) The implicit non-parametric form of an equation is, (x – xc )2 + (y – yc )2 = r2 This is an Implicit non-parametric equation In this equation, no distinction is made between the dependent and the independent variables.

  8. Parametric Equations: Parametric equations describe the dependent and independent variables in terms of a parameter. • Each point on a curve is expressed as a funcion of a parameter ‘U’. E.g.: Circle: x = r cosθ, y = r sinθ Where, θ is the parameter.

  9. Parametric curves can be defined in a constrained period (0 ≤ u ≤ 1) • The equation can be converted to a non-parametric form, by eliminating the dependent and independent variables from the equation. • Since curves are usually bounded in computer graphics, this characteristic is of considerable importance.

  10. CAD programs prefer a parametric equation for generating a curve for the following reasons: Parametric equations are converted into matrix equations – to facilitate a computer solution. Parametric equations allow great versatility in constructing space curves that are multi-valued and easily manipulated, by varying a parameter from 0 to 1 creates the points or curves. (multi valued functions)

  11. Reasons (Contd.) 3. Each point on the curve is expressed as a function of the parameter ‘U’ and this parameter acts as the local coordinate of all points on the curve. 4. Easy to express as vectors. 5. Plotting and digitizing is easy. 6. Slopes can be handled. (With non-parametric eqns., curves with slopes almost-vertical or vertical, lead to computational and programming errors) 7. More DOF can be provided.

  12. Two Categories of Parametric curves: • Analytic Curves: These are described by analytic eqns. E.g. Lines, Circles & Conics Properties: • Has a compact form which represents shapes. • Simplifies computation of related properties like areas & volumes • However, not attractive to deal with interactively.

  13. Synthetic Curves: Ones that are described by a set of data points or ‘control points’, such as Splines and Bezier Curves. Parametric eqns. usually fit these control points. Properties: • Provides designers with greater flexibility and control of curve shapes by changing the positions of the control points.

  14. Requirements for the design of curved shapes: • It should be possible to represent the shape mathematically. • The modeling should involve minimum computation. • It should be possible to generate a CNC program to machine the surfaces in 2,3,4 or 5 axes machining, or prepare a mould or die to make the part – or plastic moulding or casting.

  15. Analytic Curves: Curves can be represented by parametric eqns. presented in vector forms. E.g. lines, arc, circles, etc. • Vector form is used as it is consistent in treating 2D and 3D curves which are easy to program.

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