Geometric modeling
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This document discusses different methods for representing curves in geometric modeling, including non-parametric and parametric representations. Non-parametric representations can be explicit, using equations relating x, y, and z, or implicit, with no distinction between dependent and independent variables. Parametric representations describe variables in terms of a parameter, allowing for multi-valued and flexible curves. Specific curve types discussed include Bezier curves, defined by control points, and B-spline curves, which separate the polynomial order from the number of control points.
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This document summarizes key points from a lecture on 2D image transformations:
- 2D image transformations can be classified as either changing pixel values (filtering) or changing pixel locations (warping).
- Common 2D transformations discussed include translation, rotation, scaling, shearing, and affine transformations. Matrix representations of these transformations using homogeneous coordinates were presented.
- Transformations can be combined through matrix multiplication. The order of multiplication matters.
- Transformations were classified based on degrees of freedom, with affine transformations having 6 degrees and projective transformations having 8 degrees of freedom.
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curve one
This document provides an overview of different techniques for representing polygon meshes and parametric curves. It discusses explicit representations of polygon meshes using vertex and edge lists, as well as parametric cubic curves including Hermite, Bezier, and B-spline curves. It describes how each technique represents the geometry and constraints, and properties such as continuity and invariance. Key topics covered include representations of polygon meshes, Hermite curves defined by endpoints and tangents, Bezier curves defined by control points, and B-spline curves having local control and smooth joins.
C4 EDEXCEL HELP
The document provides revision notes on various mathematics topics including:
1) Binomial expansions, partial fractions, trigonometry formulas, and techniques for integration like volumes of revolution.
2) Parametric equations, vectors, vector equations, planes, and differential equations.
3) Details are given for solving problems involving these topics, such as using compound angle formulas, rewriting algebraic fractions as partial fractions, and separating variables to solve first-order differential equations.
Entity Manipulation
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wireframe.ppt
The document discusses wireframe modeling in CAD/CAM. A wireframe model represents an object with edges but no surfaces. It consists of geometric data defining point positions and connectivity data relating points as edges. Basic entities include points, lines, arcs, and circles. Analytic entities use mathematical equations while synthetic entities combine multiple curve segments. Bezier and B-spline curves allow more flexible shape control compared to analytic curves. The document also covers parametric representations of curves, properties of Bezier curves, and Hermite splines.
Chain Code.pptx
Chain code is a technique used in digital image processing for contour detection and representation. It encodes contours as a sequence of direction codes indicating the path from one pixel to the next along the boundary. Chain code provides a compact representation of shapes and is used for applications like contour matching, object recognition, and shape analysis. While offering efficient storage and computation, chain code can be sensitive to noise and may lose detail for complex contours.
Digital Differential Analyzer Line Drawing Algorithm
This document discusses algorithms for drawing straight line segments on a digital display. It describes how line segments are defined by their endpoint coordinates and how those coordinates are converted to integer pixel positions. It then explains how the slope-intercept equation can be used to calculate the slope and y-intercept of a line from its endpoints. Finally, it introduces the digital differential analyzer (DDA) algorithm, which uses incremental steps in x or y based on the slope to efficiently calculate pixel positions along the line segment.
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Geometric modeling
- 2. Introduction • Can be described by arrays of coordinate data or by analytical data • It is important entities in geometric modeling Curves Entities Analytical Entities Synthetic Entities
- 3. Representation of Curves Representation of Curves Non Parametric Parametric
- 4. Non Parametric Representation of Curve • The curve is represented as a relationship between x, y and z • There are two types of non parametric representation 1. Explicit Non Parametric Representation 2. Implicit Non Parametric Representation
- 5. Conti…. • Explicit non-Parametric Equation y = c1 + c2 x + c3 x2 + c4 x3 There is a unique single value of the dependent variable for each value of the independent variable. • Implicit non-parametric equation (x – xc)2 + (y – yc)2 = r2 No distinction is made between the dependent and the independent variables.
- 6. Parametric Equation • Describe the dependent and independent variables in terms of a parameter • Can be converted to a non-parametric form, by eliminating the dependent and independent variables from the equation • Allow great versatility in constructing space curves that are multi-valued and easily manipulated • Parametric curves can be defined in a constrained period (0 ≤ t ≤ 1) x = r cosθ, y = r sinθ
- 7. Bezier Curve • Difficult to change the shape of Hermite Cubic Spline • Defined by set of Data Points • Curve may interpolate or Extrapolate the data points
- 8. Characteristics of Bezier Curve • Shape of the Bezier curve is controlled by its defining points. Tangent vectors are not used in the development of curve as in case of Cubic spline • The order or Degree of Bezier curve is variable and is related to number of points defining it. • (n+1)th points define nth degree curve, which permits higher order continuity. • Data points of Bezier curve are called as Control Points
- 9. Bezier Curve
- 10. Conti……
- 11. Cubic Bezier Curve for various Control Points
- 12. B Spline Curves • Another Method to generating a curve defined by data points • It is proper and powerful generalization of Bezier Curve • Problem associate with Bezier curve is, with as increase in number of data points, the order of polynomial representing the curve is increases. • B Spline curve separates the order of polynomial representing the curve from number of given data points
- 13. Advantage • B spline curve allows local control over the shape of curve as against the global control in Bezier curve • Degree of polynomial representing the curve can be set independently of number of control points • B Spline curves gives better control • It permits add or delete any number of control of data point without changing the degree of polynomial