Geometric modeling is a part of computational geometry and applied mathematics that studies algorithms and techniques for the mathematical description of shapes.
The shapes defined in geometric modeling are generally 2D or 3D, even though several of its principles and tools can be used to sets of any finite dimension. Geometric modeling is created with computer based applications. 2D models are significant in computer technical drawing and typography. 3D models are fundamental to CAD and CAM and extensively used in many applied technical branches such as civil engineering and mechanical engineering and medical image processing.
Geometric models are commonly differentiated from object oriented models and procedural, which describe the shape perfectly by an opaque algorithm that creates its appearance. They are also compared with volumetric models and digital images which shows the shape as a subset of a regular partition of space; and with fractal models that provide an infinitely recursive description of the shape. Though, these differences are often fuzzy: for example, a image can be interpreted as a collection of colored squares; and geometric shape of circles are defined by implicit mathematical equations. Also, a fractal model gives a parametric model when its recursive description is truncated to a finite depth.
Representation of curves
A curve is an entity related to a line but which is not required to be straight. A curve is
a topological space which is internally homeomorphism to a line; this shows that a curve is a set of points which close to each of its points looks like a line, up to a deformation.
A conic section is a curve created as the intersection of a cone with a plane. In analytic geometry, a conic may be described as a plane algebraic curve of degree two, and as a quadric of dimension two.
There are several of added geometric definitions possible. One of the most practical, in that it involves only the plane, is that a non circular conic has those points whose distances to various point, called a ‘focus’,and several line, called a ‘directrix’,are in a fixed ratio, called the ‘eccentricity’.
Conventionally, the three kinds of conic section are the hyperbola, the ellipse and the parabola. The circle is a unique case of the ellipse, and is of adequate interest in its own right that it is sometimes described the fourth kind of conic section. The method of a conic relates to its ‘eccentricity’,those with eccentricity less than one is ellipses, those with eccentricity equal to one is parabolas, and those with eccentricity greater than one is hyperbolas. In the focus, directrix describes a conic the circle is a limiting with eccentricity zero. In modern geometry some degenerate methods, such as the combination of two lines, are integrated as conics as well.
Fig.2.1. Conic sections
The three kinds of conic sections are the ellipse, parabola, and hyperbola. The circle can be taken as a fourth kind of ellipse. The circle and the ellipse occur when the intersection of plane and cone is a closed curve. The circle is generated when the cutting plane is parallel to the generating of the cone. If the cutting plane is parallel to accurately one generating line of the cone, then the conic is unbounded and is mentioned a parabola. In the other case, the figure is a hyperbola.
Different factors are connected with a conic section, as shown in the Table 2.1. For the ellipse, the table shows the case of ‘a’> ‘b’, for which the major axis is horizontal; for the other case, interchange the symbols ‘a’and ‘b’.For the hyperbola the east-west opening case is specified. In all cases, ‘a’nd ‘b’are positive.
Table 2.1. Conic Sections
The non-circular conic sections are accurately those curves that, for a point ‘F’,a line ‘L’not having ‘F’and a number ‘e’ which is non-negative, are the locus of points whose distance to ‘F’equals ‘e’ multiplies their distance to ‘L’.‘F’is called the focus, ‘L’the directrix, and ‘e’the eccentricity.
i. Linear eccentricity (c) is the space between the center and the focus.
ii.Latus rectum (2l) is parallel to the directrix and passing via the focus.
iii. Semi-latus rectum (l) is half the latus rectum.
iv. Focal parameter (p) is the distance from the focus to the directrix. The relationship for the above : p*e = l and a*e=c.
A Hermite curve is a spline where every piece is a third degree polynomial defined in Hermite form: that is, by its values and initial derivatives at the end points of the equivalent domain interval. Cubic Hermite splines are normally used for interpolation of numeric values defined at certain dispute values x1,x2,x3, …..xn,to achieve a smooth continuous function. The data should have the preferred function value and derivative at each Xk. The Hermite formula is used to every interval (Xk,
Xk+1) individually. The resulting spline become continuous and will have first derivative.
Cubic polynomial splines are specially used in computer geometric modeling to attain curves that pass via defined points of the plane in 3D space. In these purposes, each coordinate of the plane is individually interpolated by a cubic spline function of a divided parameter‘t’.
Cubic splines can be completed to functions of different parameters, in several ways. Bicubic splines are frequently used to interpolate data on a common rectangular grid, such as pixel values in a digital picture. Bicubic surface patches, described by three bicubic splines, are an necessary tool in computer graphics. Hermite curves are simple to calculate but also more powerful. They are used to well interpolate between key points.
Fig.2.2. Hermite curve
The following vectors needs to compute a Hermite curve:
· P1: the start point of the Hermite curve
· T1: the tangent to the start point
· P2: the endpoint of the Hermite curve
· T2: the tangent to the endpoint
These four vectors are basically multiplied with four Hermite basis functions h1(s), h2(s), h3(s) and,h4(s) and added together.
h1(s) = 2s3 - 3s2 + 1 h2(s) = -2s3 + 3s2
h3(s) = s3 - 2s2 + s h4(s) = s3 - s2
Figure 2.3 shows the functions of Hermite Curve of the 4 functions (from left to right: h1, h2, h3, h4).
Fig.2.3. Functions of Hermite curve
A closer view at functions ‘h1’and ‘h2’,the result shows that function ‘h1’starts at one and goes slowly to zero and function ‘h2’starts at zero and goes slowly to one.
At the moment, multiply the start point with function ‘h1’and the endpoint with function ‘h2’. Let s varies from zero to one to interpolate between start and endpoint of Hermite Curve. Function
‘h3’and function ‘h4’are used to the tangents in the similar way. They make confident that the Hermite curve bends in the desired direction at the start and endpoint.
Bezier curves are extensively applied in CAD to model smooth curves. As the curve is totally limited in the convex hull of its control points P0, P1,P2 & P3, the points can be graphically represented and applied to manipulate the curve logically. The control points P0 and P3 of the polygon lie on the curve (Fig.2.4.). The other two vertices described the order, derivatives and curve shape. The Bezier curve is commonly tangent to first and last vertices.
Cubic Bezier curves and Quadratic Bezier curves are very common. Higher degree Bezier curves are highly computational to evaluate. When more complex shapes are required, Bezier curves in low order are patched together to produce a composite Bezier curve. A composite Bezier curve is usually described to as a ‘path’in vector graphics standards and programs. For smoothness assurance, the control point at which two curves meet should be on the line between the two control points on both sides.
Fig.2.4. Bezier curve
A general adaptive method is recursive subdivision, in which a curve's control points are verified to view if the curve approximates a line segment to within a low tolerance. If not, the curve is further divided parametrically into two segments, 0 ≤t ≤0.5 and 0.5 ≤ ≤t1, and the same process is used recursively to each half. There are future promote differencing techniques, but more care must be taken to analyze error transmission.
Analytical methods where a Bezier is intersected with every scan line engage finding roots of cubic polynomials and having with multiple roots, so they are not often applied in practice. A Bezier curve is described by a set of control points P0 through Pn, where ‘n’is order of curve. The initial and end control points are commonly the end points of the curve; but, the intermediate control points normally do not lie on the curve.
(i) Linear Bezier curves
2.5. Linear Bezier curve
As shown in the figure 2.5, the given points P0 and P1, a linear Bezier curve is merely a straight line between those two points. The Bezier curve is represented by And it is similar to linear interpolation.
(ii) Quadratic Bezier curves
Fig.2.6. Quadratic Bezier curve
As shown in the figure 2.6, a quadratic Bezier curve is the path defined by the function B(t), given points P0, P1, and P2,
This can be interpreted as the linear interpolate of respective points on the linear Bezier curves from P0 to P1 and from P1 to P2 respectively. Reshuffle the preceding equation gives:
The derivative of the Bezier curve with respect to the value ‘t’is
From which it can be finished that the tangents to the curve at P0 and P2 intersect at P1. While
‘t’increases from zero to one, the curve departs from P0 in the direction of P1, then turns to land at P2 from the direction of P1.
The following equation is a second derivative of the Bezier curve with respect to ‘t’:
A quadratic Bezier curve is represent a parabolic segment. Since a parabola curve is a conic section, a few sources refer to quadratic Beziers as ‘conic arcs’.
(iii) Cubic Bezier curves
As shown in figure 2.7, four control points P0, P1, P2 and P3 in the higher-dimensional space describe as a Cubic Bezier curve. The curve begins at P0 going on the way to P1 and reaches at P3 coming from the direction of P2. Typically, it will not pass through control points P1 / P2, these points are only there to give directional data. The distance between P0 and P1 determines ‘how fast’and ‘how far’the curve travels towards P1 before turning towards P2.
Fig.2.7. Cubic Bezier curve
The function B Pi, Pj, Pk (t) for the quadratic Bezier curve written by points Pi, Pj, and Pk, the cubic Bezier curve can be described as a linear blending of two quadratic Bezier curves:
For several choices of P1 and P2 the Bezier curve may meet itself.
Any sequence of any four dissimilar points can be changed to a cubic Bezier curve that goes via all four points in order. Given the beginning and ending point of a few cubic Bezier curve, and the points beside the curve equivalent to t = 1/3 and t = 2/3, the control points for the original Bezier curve can be improved.
The following equation represent first derivative of the cubic Bezier curve with respect to t:
The following equation represent second derivative of the Bezier curve with respect to t:
1. Properties Bezier curve
· The Bezier curve starts at P0 and ends at Pn; this is known as ‘endpoint interpolation’property.
· The Bezier curve is a straight line when all the control points of a cure are collinear.
· The beginning of the Bezier curve is tangent to the first portion of the Bezier polygon.
· A Bezier curve can be divided at any point into two sub curves, each of which is also a Bezier curve.
· A few curves that look like simple, such as the circle, cannot be expressed accurately by a Bezier; via four piece cubic Bezier curve can similar a circle, with a maximum radial error of less than one part in a thousand (Fig.2.8).
Fig.2.8. Circular Bezier curve
· Each quadratic Bezier curve is become a cubic Bezier curve, and more commonly, each degree ‘n’Bezier curve is also a degree ‘m’curve for any m > n.
Bezier curves have the different diminishing property. A Bezier curves does not ‘ripple’more than the polygon of its control points, and may actually ‘ripple’lss than that.
· Bezier curve is similar with respect to t and (1-t). This represents that the sequence of control points defining the curve can be changes without modify of the curve shape.
· Bezier curve shape can be edited by either modifying one or more vertices of its polygon or by keeping the polygon unchanged or simplifying multiple coincident points at a vertex (Fig .2.19).
2. Construction of Bezier curves
(i) Linear curves:
Fig.2.10. Construction of linear Bezier curve
The figure 2.10 shows the function for a linear Bezier curve can be via of as describing how far B(t) is from P0 to P1 with respect to ‘t’. When t equals to 0.25, B(t) is one quarter of the way from point P0 to P1. As ‘t’varies from 0 to 1, B(t) shows a straight line from P0 to P1.
(ii) Quadratic curves
Fig.2.11. Construction of linear Quadratic curve
As shown in figure 2.11, a quadratic Bezier curves one can develop by intermediate
points Q0 and Q1 such that as ‘t’varies from 0 to 1:
· Point Q0 (t) modifying from P0 to P1 and expresses a linear Bezier curve.
· Point Q1 (t) modifying from P1 to P2 and expresses a linear Bezier curve.
· Point B (t) is interpolated linearly between Q0(t) to Q1(t) and expresses a quadratic Bezier curve.
(iii) Higher-order curves
Fig.2.12. Construction of Higher-order curve
As shown in figure 2.12, a higher-order curves one requires correspondingly higher intermediate points. For create cubic curves, intermediate points Q0, Q1, and Q2 that express as linear
Bezier curves, and points R0 and R1 that express as quadratic Bezier curves.
3. Rational Bezier curve
Fig.2.13. Rational Bezier Curve
The rational Bezier curve includes variable weights (w) to provide closer approximations to arbitrary shapes. For Rational Bezier Curve, the numerator is a weighted Bernstein form Bezier and the denominator is a weighted sum of Bernstein polynomials. Rational Bezier curves can be used to signify segments of conic sections accurately, including circular arcs (Fig.2.13).