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Geometric Fundamentals

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Integrated Computer-Aided Design in Automotive Development

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Abstract

The chapter “Geometric Fundamentals”, introduces the reader to the mathematical and geometrical concepts which form the basis of a CAD system. It starts from scratch and leads the reader through the fields of curves, surfaces, freeform techniques, interpolation, approximation, and a range of other geometrical topics. In effect, this section might also be considered a manual for standard CAD concepts. However, rather than simply listing the methods and algorithms, this chapter actually explains the ideas behind these elements. A proper understanding of these ideas and properties can help engineers perform their jobs more effectively.

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Notes

  1. 1.

    For an application of affine transformations in CAD see also p. 267.

  2. 2.

    Throughout this text we denote the dot-product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) by \(\langle \mathbf{a}, \mathbf{b} \rangle \).

  3. 3.

    Hermann Günther Grassmann (1809–1877) was a German mathematician and philologist. He is renowned as the founder of vector and tensor calculus. Felix Christian Klein (1849–1925) was one of the most prolific and significant German mathematicians of his time, particularly in the field of geometry.

  4. 4.

    For the notion of a vector space see for instance [4].

  5. 5.

    The \(j\)th derivative of a vector function p \((t)\) with respect to \(t\) is written as \(\frac{d^j\mathbf{p}}{(dt)^j}\). If the variable is unambiguous we shortly write \(\mathbf{p}^{(j)}\) or \(\mathbf{{{{p}}}}, {{\mathbf{p}}},\) , \(\dots \).

  6. 6.

    Of course, this automatically implies that all derivatives of order \(<k\) also exist.

  7. 7.

    As for the concept of curvature see Sect. 3.3.5, p. 77

  8. 8.

    For more details we refer to ([5], p. 26).

  9. 9.

    Jean Frédéric Frenet (1816–1900) was a French mathematician who published these formulae which were also found by Joseph Alfred Serret (1819–1885).

  10. 10.

    Named after the French mathematician Étienne Bézout (1730–1783). It was first stated 1779 in É. Bézout’s Théorie générale des équations algébriques.

  11. 11.

    \(\mathrm{{gcd}}\) denotes the greatest common divisor of polynomials; see Definition 3.8, p. 67.

  12. 12.

    The point \(\mathbf{c}_{i,1}\) is constructed by subdivision of the segment by the prescribed ratio \(t:(1-t)\).

  13. 13.

    The properties of B-splines were essentially known in the nineteenth century. One of the first mathematicians to investigate the properties of such spline functions was J.C.F. Haase (1870, see [13]). The introduction of B-spline curves dates back to the Romanian mathematician I. Schoenberg (1903–1990). The theory of approximation by B-splines and algorithms for working with them are—among others—owed to the German-American mathematician Carl de Boor (born 1937 in Stolp, today Slupsk, Poland).

  14. 14.

    As for C.R. de Boor see p. 107.

  15. 15.

    Bézier and B-spline curves are defined by basis functions with this handy property. So rational Bézier or rational B-spline curves comply with this rule.

  16. 16.

    Note that any rational curve of degree \(2\) is necessarily planar.

  17. 17.

    This means that \({\fancyscript{F}}\) has the structure of a vector space: With \(F_1(t), F_2(t) \in {\fancyscript{F}}\) and \(\lambda \in \mathbb{R }\) the functions \(F_1(t) + F_2(t)\) and \(\lambda \cdot F_1(t)\) also belong to \({\fancyscript{F}}\). The set \({\fancyscript{F}}\) is closed with respect to sums and scalar multiples.

  18. 18.

    This is equivalent to the existence of a basis \(\{F_0(t), \dots , F_n(t) \}\) consisting of \(n+1\) elements.

  19. 19.

    Named after the Russian mathematician Pafnuty Lvovich Chebyshev (1821–1894). Sometimes these spaces are also called Haar spaces after the Hungarian mathematician Alfred Haar (1885–1933).

  20. 20.

    Of course, one has to interprete Eq. (3.104) for every coordinate of the vectors \(\mathbf{a}_i\), \(\mathbf{c}_i\) separately!

  21. 21.

    Joseph-Louis Lagrange (1736–1813) was a French mathematician who also contributed to analytical mechanics and mathematical physics.

  22. 22.

    This algorithm is due to Alexander Craig Aitken (1895–1967), a mathematician from New Zealand.

  23. 23.

    Albert W. Overhauser, born 1925 in San Diego, California is an American physicist.

  24. 24.

    Charles Hermite (1822–1901) was a French mathematician who particularly contributed to the fields of number theory and algebra.

  25. 25.

    Equation (3.123) has to be interpreted for each coordinate of the vectors \(\mathbf{t}_i\), \(\mathbf{c}_i\) separately, so it basically consists of three equations.

  26. 26.

    The inverse of a tridiagonal matrix can for instance be computed by the method suggested in [17] though we do not recommend going this way. Instead we opt for the numerically stable Gauss-Jordan method with pivoting for solving the linear equation system (3.123) which is a standard mathematical tool.

  27. 27.

    Remember the similar situation with the Hermite approach to the same task.

  28. 28.

    In fact its invertibility is easy to check after expanding the matrix along the 2nd and the last but one row. The remaining matrix is main diagonal dominant (see p. 130) and hence its determinant is non-zero.

  29. 29.

    The same thing holds for the first Interpolation Problem 3.1. If we interpret the scalar values \(c_i\) as ‘points’ on the real number line we can apply the following adaption of the parameter sequence \(s_0, \dots , s_n\) correspondingly.

  30. 30.

    Usually, approximation deals with large data sets: \(m > n\); the case \(m\le n\) would offer the opportunity to switch over to an interpolation task (see Sect. 3.5).

  31. 31.

    A matrix \(\mathbf{M}\) of dimension \((n+1)\times (n+1)\) is called positive semidefinite if

    \([x_0, \dots , x_n] \cdot \mathbf{M} \cdot \left[ \begin{array}{c}x_0 \\ \vdots \\ x_n\end{array}\right] \ge 0\) for all \(x_0, \dots , x_n \in \mathbb{R }\).

  32. 32.

    Of course, the approximation task for points can also be solved ‘coordinate-wise’ with the help of Proposition 3.14 We still want to note Proposition 3.15 explicitely as a condensed account on the frequently occurring task regarding points.

  33. 33.

    This equation has to be interpreted separately for each component of the vectors \(\mathbf{a}_i\), \(\mathbf{c}_j\).

  34. 34.

    Here, dots indicate differentiation with respect to the parameter \(t\) on the curve.

  35. 35.

    Mind that the vector space of cubic splines defined on a given knot vector is—in general—not a Chebyshev space (compare Proposition 3.14, p. 138)!

  36. 36.

    Of course, this automatically implies that all derivatives of order \(<k\) exist as well.

  37. 37.

    This is the so-called geographic parameterization of a sphere as the parameters \(u\) and \(v\) represent the geographic longitude and latitude, respectively.

  38. 38.

    gcd denotes the greatest common divisor of polynomials; see Definition 3.12, p. 69.

  39. 39.

    For the notion of ruled surfaces see Sect. 3.7.9.

  40. 40.

    It can be proved: If the tangent planes of two distinct points on a generator of a ruled surface are identical, any other point on this generator will also have this tangent plane.

  41. 41.

    This is the kind of surface which can be machined on a lathe.

  42. 42.

    Here, reversing the net refers to swapping one index—say \(i\)—from \(i\) to \((m-i)\) and leaving the other index untouched. Reversing the net with respect to the second index—here, replacing \(j\) by \((n-j)\)—is a different operation. Both actions combined make for 4 options overall.

  43. 43.

    The choice of such control points is the job of the stylist. The expression deliberately chosen is only meant in terms of geometric correctness.

  44. 44.

    As for the term spline function and subspline function see Definition 3.28, p. 99.

  45. 45.

    A kinematic surface is generated by subjecting a curve to a Euclidean motion.

  46. 46.

    The blending functions are the Bernstein polynomials of degree \(1\); the tensor product surface is a \((1,1)\)-Bézier surface.

  47. 47.

    Steven Anson Coons (1912–1979) was an early pioneer in the field of computer graphics at the MIT (Massachusetts Institute of Technology).

  48. 48.

    The term rectangular refers to a matrix of points \(\mathbf{c}_{i,j}\), \(i = 0, \dots , m\), \(j = 0, \dots , n\).

  49. 49.

    The reader might as well imagine the functions \(F_i(u)\) and \(G_j(v)\) to be the Bernstein polynomials of degree \(m\) and \(n\), respectively.

  50. 50.

    Of course, this equation has to be interpreted for each component of the vectors \(\mathbf{a}_{i,j}\), \(\mathbf{c}_{i,j}\) separately.

  51. 51.

    This is for instance the case if both, \({\fancyscript{F}}\) and \({\fancyscript{G}}\), are Chebyshev spaces; see Definition 3.36, p. 117.

  52. 52.

    We say that a surface fits a given set of points, if certain distances between the points and some surface points are minimized. At the moment this explanation may seem faint, but it will be substantiated further on.

  53. 53.

    If \(\mathbf{E}\) was the identity matrix, we would have an ordinary eigenvalue problem as opposed to the case where \(\mathbf{E}\) is an arbitrary matrix; then the eigenvalue problem is called generalized.

  54. 54.

    Gerolamo Cardano (1501–1576) was an Italian Renaissance mathematician. Cardano’s formula for the solution of \(3^{rd}\) order equations can be found in every formulary booklet.

  55. 55.

    Here we adapt the method for parameter correction which we have already applied in the univariate case; see Sect. 3.6, p. 136.

  56. 56.

    Subscribed \(u\)’s and \(v\)’s indicate differentiation with respect to to \(u\) and \(v\).

  57. 57.

    For modeling with solids see also Sect. 4.3, p. 276.

  58. 58.

    Note that what we call the roofline is actually the boundary curve of the window towards the roof plus its boundary curve towards the A-pillar of the car.

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  1. Institute of Automotive Engineering, Graz University of Technology, Inffeldgasse 11/1, 8010, Graz, Austria

    Mario Hirz

  2. MAGNA STEYR Engineering AG & Co KG, Liebenauer Hauptstr. 317, 8041, Graz, Austria

    Wilhelm Dietrich

  3. Institute of Geometry, Graz University of Technology, Kopernikusgasse 24/IV, 8010, Graz, Austria

    Anton Gfrerrer & Johann Lang

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Hirz, M., Dietrich, W., Gfrerrer, A., Lang, J. (2013). Geometric Fundamentals. In: Integrated Computer-Aided Design in Automotive Development. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11940-8_3

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