Parametric Equations

5 Key excerpts on "Parametric Equations"

• 

  eBook - ePub

  Geometry for Naval Architects

  • Adrian Biran(Author)
  • 2018(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 4

  Parametric Curves

  Abstract

  A curve can be represented by an implicit equation,

  f ( x , y ) = 0

  ,

  f ( x , y , z ) = 0

  , by an explicit equation,

  y = f ( x )

  ,

  z = f ( x , y )

  , or by Parametric Equations,

  x =

  f 1

  ( t )

  ,

  y =

  f 2

  ( t )

  ,

  z =

  f 3

  ( t )

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  3D Graphics for Game Programming

  • JungHyun Han(Author)
  • 2011(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  6.1 Parametric Curves A parametric curve is described as a function of a parameter t . A line is an instance of a curve. Section 6.1.1 starts with presenting a parameterized line segment and extends it into quadratic and cubic B´ ezier curves. Section 6.1.2 presents another type of parametric curve, the Hermite curve. 1 There are numerous books in the area of parametric curves and surfaces. An excellent introductory book is [22]. 131 132 3D Graphics for Game Programming 6.1.1 B´ ezier Curves Fig. 6.1 : A line segment connecting two end points is represented as a linear interpolation of the two points. Consider a line segment bounded by two points p 0 and p 1 , shown in Fig. 6.1. It can be represented in a parametric equation: p ( t ) = (1 -t ) p 0 + tp 1 (6.1) where t is in the range [0,1]. The function p ( t ) maps an interval of real values represented by parameter t to a continuous set of points. Suppose that t takes a specific value. Then, p ( t ) in Equation (6.1) is reduced to a point. It corresponds to a linear interpolation of p 0 and p 1 , and can be described as a weighted sum of the two points. The line segment is divided into two parts by p ( t ), and the weight for an end point is proportional to the length of the part “on the opposite side,” i.e., the weights for p 0 and p 1 are (1 -t ) and t , respectively, in Fig. 6.1. The function p ( t ) is vector-valued, e.g., p ( t ) = ( x ( t ) , y ( t ) , z ( t )) for 3D space. When p 0 = ( x 0 , y 0 , z 0 ) and p 1 = ( x 1 , y 1 , z 1 ), linear interpolation is applied to each of the xyz -components: p ( t ) =   x ( t ) y ( t ) z ( t )   =   (1 -t ) x 0 + tx 1 (1 -t ) y 0 + ty 1 (1 -t ) z 0 + tz 1   Two points define a line segment, not a general curve. For defining a curve, we need three or more points. A well-known technique for constructing a parametric curve using those points is the de Casteljau algorithm , named after its inventor Paul de Casteljau, a French mathematician at Citro¨ en.

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  Graphics and Visualization

  Principles & Algorithms

  • T. Theoharis, Georgios Papaioannou, Nikolaos Platis, Nicholas M. Patrikalakis(Authors)
  • 2008(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  7 Parametric Curves and Surfaces Equations are just the boring part of mathematics. I attempt to see things in terms of geometry. —Stephen Hawking 7.1 Introduction In Chapter 2 we presented algorithms for the rasterization of basic geometric primitives, lines and circles. However, the composition of realistic graphics scenes calls for more flexible, free-form curves and surfaces. The area of computer graphics that deals with these shapes is computer-aided geometric design (CAGD). In this chapter we shall examine representations and properties of the most basic forms of such curves and surfaces; the reader should refer to [Fari01, Hosc96, Bart87] for more advanced topics. The need for mathematical representations of free-form shapes, suitable for computer processing, became apparent during the 1960s in the automotive and aeronautic industries. Until that time, the specifications by the designers for the shape of cars and planes were implemented only approximately, as no exact de-scriptions of such shapes were in practical use. When computer-driven machinery that could produce complex-shaped objects was made available to these indus-tries, it became essential to devise suitable mathematical descriptions. Paul de Casteljau and Pierre B´ ezier, then working at Citro¨ en and Renault, respectively, developed independently the theory of polynomial curves and surfaces that now bears B´ ezier’s name—de Casteljau’s work was not published early on—and con-stitutes the basic tool for describing and rendering free-form shapes. 191 192 7. Parametric Curves and Surfaces All of the curve and surface descriptions examined in the rest of this Chap-ter are in parametric form, and the reader is referred to Appendix B (especially Sections B.1.1 and B.2.1) for an overview of the relevant background theory.

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  Mathematical Techniques in GIS

  • Peter Dale(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  181 9 Curves and Surfaces 9.1 PARAMETRIC FORMS So far, we have seen that for points on a plane we can express a straight line as a series of values ( x , y ) such that ax + by + c = 0 We can also express it in the form that y = mx + n where m is the slope or gradient of the line and n is a constant. These two expressions represent the same line if m = – a/b and n = – c/b We also saw in Chapter 6 and can see from Figure 9.1 that if the slope of the line AB is θ (measured from the horizontal axis anticlockwise), then y = x tan θ + c Here, c is the value of y when x is zero. We can also express this as d y /d x = tan θ = m , which is constant for a straight line since ∫ tan θ d x = x tan θ + c The differential of a constant is zero; hence, for a line, d 2 y /d x 2 = 0. Also note that if we use clockwise bearings from the vertical rather than anti-clockwise angles from the horizontal, the bearing of line AB = (90 – θ ) and y = x cot (Bearing AB ) + c If the line CD is perpendicular to AB, its slope will be (90 + θ ). Hence, for the line CD y = m ′ x + c ′ where m ′ = –cot ( θ ) = –tan (Bearing AB ) 182 Mathematical Techniques in GIS We have also seen that when two lines AB and CD are at right angles, their slopes are such that m m ′ = tan θ * (–cot θ ) = –1 If the product of the slopes of two lines = –1 they are said to be orthogonal . Thus, in Figure 9.1, the lines AB and CD are orthogonal. In Chapter 7, we also used the term orthogonal for a matrix where A A –1 = I . In the form y = mx + n , y is the dependent variable and x the independent. We can however express a straight line in parametric form as x = p + t cos θ ; y = q + t sin θ where ( p , q ) are the coordinates of a point on the line, θ is the fixed slope angle, and “ t ” is the parameter. Here, t represents the distance along the line from point ( p , q ). There is then only one independent variable ( t ) and two dependent variables ( x and y) .

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  Mathematics for the Nonmathematician

  • Morris Kline(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  t = 1, x = 3, and y = 9. Then (3 , 9) are the coordinates of a point on the curve, namely, the point A , which we discussed earlier. For , x = 4 and y = 16, and (4 , 16) are the coordinates of the point B .

  We may also say that the two formulas x = 3t and y = 9t 2 are equivalent to the single formula y = x 2 . Whether we speak of equations of curves or formulas is really immaterial. The word formula emphasizes the idea of change because formulas are relationships among variables, and we often like to think of what happens to one variable as another, related variable changes. On the other hand, when a curve is given in its entirety, the concept of change may not be relevant, and then we speak of the equation of the curve.

  If the two formulas in (1) are entirely equivalent to the single formula y = x 2 , why do we bother with two formulas instead of one? There are two reasons: (1) When one argues from physical principles, it is often easier to arrive at the parametric representation of a given phenomenon, and (2) it is easier to study the phenomenon by working with Parametric Equations. We shall recognize the utility of parametric representations as we study the next few sections.

  There is one more mathematical detail. Suppose that we find the parametric formulas describing a motion and we wish to determine the direct relationship between x and y . Can we do this? Yes indeed. For example, if x = 3t and y = 4t 2 are the parametric formulas, we can solve the first one for t and obtain t = x /3. We substitute this value of t in y = 4t 2

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