Central Pole-to-Plane Projections on Ellipsoids and Elliptic Paraboloids: Geometry and Applications

(NOTE: Every chapter ends with Questions to Guide Your Review, Practice Exercises, and Additional Exercises.) P. Preliminaries. Real Numbers and the Real Line. Coordinates, Lines, and Increments. Functions. Shifting Graphs. Trigonometric Functions. 1. Limits and Continuity. Rates of Change and Limits. Rules for Finding Limits. Target Values and Formal Definitions of Limits. Extensions of the Limit Concept. Continuity. Tangent Lines. 2. Derivatives. The Derivative of a Function. Differentiation Rules. Rates of Change. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation and Rational Exponents. Related Rates of Change. 3. Applications of Derivatives. Extreme Values of Functions. The Mean Value Theorem. The First Derivative Test for Local Extreme Values. Graphing with y e and y . Limits as x a a, Asymptotes, and Dominant Terms. Optimization. Linearization and Differentials. Newton's Method. 4. Integration. Indefinite Integrals. Differential Equations, Initial Value Problems, and Mathematical Modeling. Integration by Substitution--Running the Chain Rule Backward. Estimating with Finite Sums. Riemann Sums and Definite Integrals. Properties, Area, and the Mean Value Theorem. Substitution in Definite Integrals. Numerical Integration. 5. Applications of Integrals. Areas Between Curves. Finding Volumes by Slicing. Volumes of Solids of Revolution--Disks and Washers. Cylindrical Shells. Lengths of Plan Curves. Areas of Surfaces of Revolution. Moments and Centers of Mass. Work. Fluid Pressures and Forces. The Basic Pattern and Other Modeling Applications. 6. Transcendental Functions. Inverse Functions and Their Derivatives. Natural Logarithms. The Exponential Function. ax and logax. Growth and Decay. L'Hopital's Rule. Relative Rates of Growth. Inverse Trigonomic Functions. Derivatives of Inverse Trigonometric Functions Integrals. Hyperbolic Functions. First Order Differential Equations. Euler's Numerical Method Slope Fields. 7. Techniques of Integration. Basic Integration Formulas. Integration by Parts. Partial Fractions. Trigonometric Substitutions. Integral Tables and CAS. Improper Integrals. 8. Infinite Series. Limits of Sequences of Numbers. Theorems for Calculating Limits of Sequences. Infinite Series. The Integral Test for Series of Nonnegative Terms. Comparison Tests for Series of Nonnegative Terms. The Ratio and Root Tests for Series of Nonnegative Terms. Alternating Series, Absolute and Conditional Convergence. Power Series. Taylor and Maclaurin Series. Convergence of Taylor Series Error Estimates. Applications of Power Series. 9. Conic Sections, Parametrized Curves, and Polar Coordinates. Conic Sections and Quadratic Equations. Classifying Conic Sections by Eccentricity. Quadratic Equations and Rotations. Parametrizations of Plan Curves. Calculus with Parametrized Curves. Polar Coordinates. Graphing in Polar Coordinates. Polar Equations for Conic Sections. Integration in Polar Coordinates. 10. Vectors and Analytic Geometry in Space. Vectors in the Plane. Cartesian (Rectangular) Coordinates and Vectors in Space. Dot Products. Cross Products. Lines and Planes in Space. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates. 11. Vector-Valued Functions and Motion in Space. Vector-Valued Functions and Space Curves. Modeling Projectile Motion. Arc Length and the Unit Tangent Vector T. Curvature, Torison, and the TNB Frame. Planetary Motion and Satellites. 12. Multivariable Functions and Partial Derivatives. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Differentiability, Linearization, and Differentials. The Chain Rule. Partial Derivatives with Constrained Variables. Directional Derivatives, Gradient Vectors, and Tangent Planes. Extreme Values and Saddle Points. Lagrange Multipliers. Taylor's Formula. 13. Multiple Integrals. Double Integrals. Areas, Moments, and Centers of Mass. Double Integrals in Polar Form. Triple Integrals in Rectangular Coordinates. Masses and Moments in Three Dimensions. Triple Integrals in Cylindrical and Spherical Coordinates. Substitutions in Multiple Integrals. 14. Integration in Vector Fields. Line Integrals. Vector Fields, Work, Circulation, and Flux. Path Independence, Potential Functions, and Conservative Fields. Green's Theorem in the Plane. Surface Area and Surface Integrals. Parametrized Surfaces. Stokes's Theorem. The Divergence Theorem and a Unified Theory. Appendices. Mathematical Induction. Proofs of Limit Theorems in Section 1.2. Complex Numbers. Simpson's One-Third Rule. Cauchy's Mean Value Theorem and the Stronger Form of L'Hopital's Rule. Limits that Arise Frequently. The Distributive Law for Vector Cross Products. Determinants and Cramer's Rule. Euler's Theorem and the Increment Theorem.

Approximation of granular textures by quadric surfaces

Modern architecture of civil engineering structures made of reinforced concrete is demanding solutions to problems related to design and construction of roofs shaped as creative forms which require an adequate combination of arts and physical-mathematical models. In that sense, the quadric and cylindrical surfaces offer an attractive based on their volumetric expression, historicity, and mathematical feasibility. In this work, seven cylindrical and quadric surfaces named elliptic cylinder, parabolic cylinder, elliptic cone, elliptic paraboloid, hyperbolic paraboloid, ellipsoid and hyperboloid of two sheets have been analysed to find out which of them show to be more efficient when used as reinforced concrete roof structures. To do so, the configuration of each structure hase been adjusted in terms of similar stiffness, strength parameters and enclosed volume. The structural efficiency was computed using the results of weight/load ratio, available strength, material consumption and relative stiffness. The solution of the model has been achieved using a combination of exact solutions and numerical methods. To compute the model results a Matlab® code was written and validated using the structural software SAP2000®. The best structural efficiency in terms of stiffness, strength, and materials consumption was obtained for roofs configurated using surfaces built from parabolas, i.e., elliptic paraboloid, hyperbolic paraboloid, and parabolic cylinder. The results of this work can be used in future studies and applications related to arhitectural comfort, structural behaviour and material consumption in construction of roofs.

In this paper, an analytical method based on Timoshenko theory is derived for obtaining the in-plane static closed-form general solutions of deep curved laminated piezoelectric beams with variable curvatures. The equivalent modulus of elasticity is utilized to take into account the material couplings in the laminated beam. The linear piezoelectric effect is considered to develop the static governing equations. The governing differential equations are formulated as functions of the angle of tangent slope by introducing the coordinate system defined by the arc length of the centroidal axis and the angle of tangent slope. To solve the governing equations, defined are the fundamental geometric properties, such as the moments of the arc length with respect to horizontal and vertical axes. As the radius is known, the fundamental geometric quantities can be calculated to obtain the static closed-form solutions of the axial force, shear force, bending moment, rotation angle, and displacement fields at any cross-section of curved beams. The closed-form solutions of the circle beams covered with piezoelectric layers under various loading cases are presented. The results show the consistency in comparison with finite results. Solutions of the non-dimensional displacements for the laminated circular and spiral curved beams with different lay-ups are available. The non-dimensional displacements with geometry and material parameters are also investigated.

Chapter 2 - Geometric Fundamentals

This thesis presents a theory of multi-scale, curvature and torsion based shape representation for planar and space curves. The theory presented has been developed to satisfy various criteria considered useful for evaluating shape representation methods in computer vision. The criteria are: invariance, uniqueness, stability, efficiency, ease of implementation and computation of shape properties. The regular representation for planar curves is referred to as the curvature scale space image and the regular representation for space curves is referred to as the torsion scale space image. Two variants of the regular representations, referred to as the renormalized and resampled curvature and torsion scale space images, have also been proposed. A number of experiments have been carried out on the representations which show that they are very stable under severe noise conditions and very useful for tasks which call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation. \n Planar or space curves are described at varying levels of detail by convolving their parametric representations with Gaussian functions of varying standard deviations. The curvature or torsion of each such curve is then computed using mathematical equations which express curvature and torsion in terms of the convolutions of derivatives of Gaussian functions and parametric representations of the input curves. Curvature or torsion zero-crossing points of those curves are then located and combined to form one of the representations mentioned above. \n The process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve. This thesis contains a number of theorems about evolution and arc length evolution of planar and space curves along with their proofs. Some of these theorems demonstrate that evolution and arc length evolution do not change the physical interpretation of curves as object boundaries and others are in fact statements on the global properties of planar and space curves during evolution and arc length evolution and their representations. Other theoretical results shed light on the local behavior of planar and space curves just before and just after the formation of a cusp point during evolution and arc length evolution. Together these results provide a sound theoretical foundation for the representation methods proposed in this thesis.

In this paper we address the shape completion problem, i.e., the geometric continuation of boundaries of objects which are temporarily interrupted by occlusion. Also known as the gap completion or curve completion problem, this problem is a significant element of perceptual grouping of edge elements and has been approached by using cubic splines or biarcs which minimize total curvature squared ( elastica), as motivated by a physical analogy.Our approach is motivated by railroad design methods used in the early 1900’s by civil engineers to connect two rail segments by “transition curves”, and by the work of Knuth on mathematical typography. We propose that in using an energy minimizing solution shape completion curves should not penalize curvature but curvature variation. The minimization of total curvature variation leads to an Euler Spiral solution whose curvature varies linearly with arc length. We examine the construction of this curve using a nonlinear system of equations involving Fresnel Integrals, whose solution relies on optimization from a suitable initial condition constrained to satisfy given boundary conditions. Since the choice of an appropriate initial curve is critical in this optimization, we analytically derive an optimal solution in the class of biarc curves, which is then used as the initial curve. The resulting interpolations yield intuitive interpolation across gaps and occlusions and are, in contrast to scale invariant elastica, extensible. In addition, Euler Spiral segments can be used to model boundary segments between curvature extrema and to model skeletal branch geometry.Keywordsperceptual groupingshape completiongap completionedge salienceEuler Spiralintrinsic splinesbiarcsinterpolationboundary modeling

This paper proposes an image-based approach to reconstruct the shape of continuum biological (e.g., tendrils of climbing plants) and artificial (continuum soft robots) structures which can deform into coils or curls with variable curvature that depends on the arc length. The proposed method is based on 2D clothoid curves, for which we explore two resolution approaches: i) single-segment clothoid representation, with optimal curve parameters search; ii) piece-wise clothoid representation, with G1 Hermite-fitting solution. Besides, we propose a novel algorithm to sort 2D unarranged points that addresses the issue of possible undesired branches and discontinuities. We numerically evaluate the performance of the method and compare it with a constant curvature fitting. We obtain an improvement of more than 100% in tendril and up to 5% with a soft continuum robotic artefact, demonstrating the feasibility and the reliability of our approach. The proposed model can be applied for shape representation and reconstruction on both long slender living organisms and continuum soft robots with curling-like behavior.

The paper investigates self-intersections of offsets of implicit quadratic surfaces. The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can be locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e. the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.

This letter provides a tighter piecewise linear approximation of generating units' quadratic cost curves (QCCs) for unit commitment (UC) problems. In order to facilitate the UC optimization process with efficient mixed-integer linear programing (MILP) solvers, QCCs are piecewise linearized for converting the original mixed-integer quadratic programming (MIQP) problem into an MILP problem. Traditionally, QCCs are piecewise linearized by evenly dividing the entire real power region into segments. This letter discusses a rigorous segment partition method for obtaining a set of optimal segment points by minimizing the difference between chord and arc lengths, in order to derive a tighter piecewise linear approximation of QCCs and, in turn, a better UC solution as compared to the equipartition method. Numerical test results show the effectiveness of the proposed method on a tighter piecewise linear approximation for better UC solutions.

This paper is primarily concerned with the mathematical formulation of the conditions for intersection of two surfaces described by general second degree polynomial (quadratic) equations. The term quadric surface is used to denote implicitly a surface described by a quadratic equation in three variables. Of special interest is the case of two ellipsoids in the three dimensional space for which the determination of intersection has practical applications. Even the simplest of traditional approaches to this intersection determination has been based on a constrained numerical optimization formulation in which a requisite combined rotational, translational and dilational transformation reduces one of the ellipsoids to a sphere, and then a numerical search procedure is performed to obtain the point on the other ellipsoid closest to the sphere's center. Intersection is then determined according to whether this shortest distance exceeds the radius of the sphere. An alternative novel technique, used by Alfano and Greer [AG01] is based on formulating the problem in four dimensions and then determining the eigenvalues which yield a degenerate quadric surface. This method has strictly relied on many numerical observations of the eigenvalues to arrive at the conclusion whether these ellipsoids intersect. A rigorous mathematical formulation and solution was provided by Chan [Cha01] to explain the myriads of numerical observations obtained through trial and error using eigenvalues. Moreover, it turns out that this mathematical analysis may also be extended in two ways. First, it is also valid for quadric surfaces in general: ellipsoids, hyperboloids of one or two sheets, elliptic paraboloids, hyperbolic paraboloids, cylinders of the elliptic, hyperbolic and parabolic types, and double elliptic cones. (The term ellipsoids includes spheres, and elliptic includes circular.) The general problem of analytically determining the intersection of any pair of these surfaces is not simple. This formulation provides a much desired simple solution. The second way of generalization is to extend it to n dimensions in which we determine the intersection of higher dimensional surfaces described by quadratic equations in n variables. The analysis using direct substitution and voluminous algebraic simplification turns out to be very laborious and troublesome, if at all possible in the general case. However, by using abstract symbolism and invariant properties of the extended (n+1) by (n+1) matrix, the analysis is greatly simplified and its overall structure made comprehensive and comprehensible. These results are also included in this paper. They also serve as a starting point for further theoretical investigations in higher dimensional analytical geometry.

The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy E, the axial component of the angular momentum, Lz, and Carter's constant Q. These parameters are obtained by solving the Hamilton–Jacobi equation for the dynamical problem of geodesic motion. Employing the action-angle variable formalism, on the other hand, yields a different set of constants of motion, namely, the fundamental frequencies ωr, ωθ and ωϕ associated with the radial, polar and azimuthal components of orbital motion, respectively. These frequencies, naturally, determine the time scales of orbital motion and, furthermore, the instantaneous gravitational wave spectrum in the adiabatic approximation. In this paper, it is shown that the fundamental frequencies are geometric invariants and explicit formulae in terms of quadratures are derived. The numerical evaluation of these formulae in the case of a rapidly rotating black hole illustrates the behaviour of the fundamental frequencies as orbital parameters, such as the semi-latus rectum p, the eccentricity e or the inclination parameter θ− are varied. The limiting cases of circular, equatorial and Keplerian motion are investigated as well and it is shown that known results are recovered from the general formulae.

In this study, the generalized offset curve of the pseudo-null curve is defined in terms of the vector elements of the original curve by using Serret Frenet and Bishop frames. The generalized curves are analyzed with a more precise approximation by using the ϵ-neighborhood approach. This method allows for the examination of variations in arc length and curvatures between the generalized offset curves and the original pseudo-null curves, using the Serret Frenet and Bishop parameters. In conclusion, illustrative examples are presented.

A stereoscopic drawing program is described which permits the user to display and manipulate quadric surfaces. The quadric surfaces are the 3-dimensional relatives of the ellipse, parabola and hyperbola and include ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids. These surfaces have both implicit and parametric representations. A 3-button mouse is used to create and manipulate the surfaces. Rubber-banding can be used to define the surface and three dimensional transformations of the surface including scaling, rotation and translation are defined by mouse movement. A goal is to maintain a consistent and intuitive method of control for these surfaces, using techniques similar to those used in 2-dimensional drawing systems. The tessellation, color and shading characteristics of a surface can be determined interactively by the user.

Computing minimum distance between two implicit algebraic surfaces