DOCSLIB.ORG

Geometry of Perspective Imaging Images of the 3-D World

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry of Perspective Imaging Images of the 3-D World Geometry of perspective imaging ■ Coordinate transformations ■ Image formation ■ Vanishing points ■ Stereo imaging Image formation Images of the 3-D world ■ What is the geometry of the image of a three dimensional object? – Given a point in space, where will we see it in an image? – Given a line segment in space, what does its image look like? – Why do the images of lines that are parallel in space appear to converge to a single point in an image? ■ How can we recover information about the 3-D world from a 2-D image? – Given a point in an image, what can we say about the location of the 3-D point in space? – Are there advantages to having more than one image in recovering 3-D information? – If we know the geometry of a 3-D object, can we locate it in space (say for a robot to pick it up) from a 2-D image? Image formation Euclidean versus projective geometry ■ Euclidean geometry describes shapes “as they are” – properties of objects that are unchanged by rigid motions » lengths » angles » parallelism ■ Projective geometry describes objects “as they appear” – lengths, angles, parallelism become “distorted” when we look at objects – mathematical model for how images of the 3D world are formed Image formation Example 1 ■ Consider a set of railroad tracks – Their actual shape: » tracks are parallel » ties are perpendicular to the tracks » ties are evenly spaced along the tracks – Their appearance » tracks converge to a point on the horizon » tracks don’t meet ties at right angles » ties become closer and closer towards the horizon Image formation Example 2 ■ Corner of a room – Actual shape » three walls meeting at right angles. Total of 270o of angle. – Appearance » a point on which three lines segments are concurrent. Total angle is 360o Image formation Example 3 q ■ B appears between A and C from point p A ■ But from point 1, A appears between B C and C ■ Apparent displacement of objects due to B change in viewing position is called parallax shift p – Greeks knew that is the earth revolved around the sun there would be parallax shift of the stars C – Tycho Brahe looked for this shift, but A his instruments were not accurate B enough. Image formation Vanishing points and lines ■ To begin with, let’s assume we’re looking at a scene on the plane, σ, which is perpendicular to the image plane ρ. – To each point P on σ we associate the v C point p on ρ corresponding to the V intersection of the line CP with the plane ρ. – C is called the center of projection p ■ The point of intersection of the line P through the optical center, C, that is perpendicular to the image plane (and so Object plane, σ parallel to the object plane) is called the principal vanishing point, V. ■ The line, v, which is the intersection of the picture plane and the plane through P Image plane, ρ parallel to the object plane is called the vanishing line, or the horizon line. Image formation Vanishing points and lines ■ Our mapping between σ and ρ does not define a 1-1 correspondence. – Points on v are not the image of any point on σ – Points on m (intersection of plane ρ through C parallel to ) have no v C image on σ V ■ We’ll fix this later. ■ Add railroad tracks to σ. p P m Object plane, σ Image plane, ρ Image formation Vanishing points and lines ■ The images of all the points on L1 lie on the π plane 1 defined by L1 and C. – this is because the lines of sight for each point on v L1 lie on this plane. C ■ π V The image of L1 is then the intersection of 1 with the picture plane - l1. ■ L1 is parallel to CV since they are both perpendicular to the picture plane. l2 l1 π L2 – But, then CV must also lie on 1, since parallel lines are co-planar, and V lies on π 1. L1 ■ So, the principal vanishing point must lie on l1, Object plane, σ π since it lies on both 1 and the picture plane. ■ Similarly, V must lie on l2, the image of L2. ■ In fact, ALL lines in the object plane that are ρ perpendicular to the image plane image to lines Image plane, passing through V. – This is the image of the “point at infinity” for that set of parallel lines Image formation Vanishing points and lines ■ M1 and M2 are parallel, but not perpendicular to the image plane. ■ The image of M is the line m where the plane, i i v π , containing C and M intersects the image Λ C i i V plane. M V ■ There is a unique line, Λ, through C which is π parallel to Mi and so is this line is also on i. m2 ■ Since this line is horizontal, it also lies in the M m1 plane determined by C and v, the vanishing line. 2 M ■ Since v and Λ lie in the same plane, and are not 1 Object plane, σ parallel, they must intersect at some point Vm. Image plane, ρ Image formation Vanishing points and lines ■ π This places Vm on both i and the image plane, so it must lie on their intersection, mi. v C – So, any family of parallel lines on the Λ V V object plane will image to a set of lines M passing through a point (vanishing point) on the vanishing line. ■ As we rotate the set of lines, the vanishing m1 m2 point moves along the vanishing line. M2 ■ So even though parallel lines don’t “meet” we M1 can see where they meet in images! Object plane, σ Image plane, ρ Image formation General case ■ Generally, the vanishing line for any object plane is the line of intersection of the –image plane – plane parallel to the object plane through C (called the horizon plane). ■ The vanishing point for a line on the object plane is the intersection of the – vanishing line – plane containing the original line and the center of projection. V C v π ρ Image formation Vanishing points and lines Image formation Homogeneous coordinates ■ Classical Euclidean geometry: through any point not on a given line, there exists a unique line which is parallel to the given line. – For 2,000 years, mathematician tried to “prove” this from Euclid’s postulates. – In the early 20’th century, geometry was revolutionized when mathematicians asked: What if this were false? – That is, what if we assumed that EVERY pair of lines intersected? – To do this, we’ll have to add points and lines to the standard Euclidean plane. ■ If (x,y) are the rectangular coordinates of a point, P, and if (x1, x2, x3) are any three real numbers such that: –x1/x3 = x –x2/x3 = y then (x1, x2, x3) are a set of homogeneous coordinates for (x,y). ■ So, in particular, (x,y,1) are a set of homogeneous coordinates for (x,y) Image formation Homogeneous coordinates ■ Given the homogeneous coordinates, (x1, x2, x3), the rectangular coordinates can be recovered. ■ But (x,y) has an infinite number of homogeneous coordinate representations, because if (x1, x2, x3) are homogeneous coordinates of (x,y), then so are (kx1, kx2,k x3) for any k <>0. Image formation Points and lines ■ In rectangular coordinates, we can represent a line by: – l1x + l2y + l3 = 0 l1, l2 not both 0. ■ In homogeneous coordinates, the equation becomes – l1x1 + l2x2 + l3x3 = 0 ■ So a line is determined by the three coefficients (l1, l2, l3) ■ If l is the line l1x1 + l2x2 + l3x3 = 0, then we say that [l1, l2, l3] are the homogeneous coordinates of l. – So, in homogeneous coordinates, points and lines look the same - triples of numbers. – We’ll represent points as (…) and lines as […] to keep things straight. ■ A line is uniquely determined by a set of homogeneous coordinates, but a given line has an infinite number of homogeneous coordinate representations. –If [l1, l2, l3] represents a given line, then so does [kl1, kl2,k l3]. » I.e., the same set of points satisfy the equations l1x1 + l2x2 + l3x3 = 0 and kl1x1 +k l2x2 + kl3x3 = 0 Image formation Vectors: Inner Product ■ T Inner product of two vectors a=(ax ay az) and b=(b b b )T is a scalar b x y z α α a·b =|a|·|b|cos = axbx + ayby+ az bz |b|cosα a Note that we are projecting one vector onto the other. ■ Length: ≡ T 1/2 2 2 2 1/2 |a| ||a|| =(a ·a) = (ax + ay + az ) Image formation Vectors: Cross product ■ Vector product of two vectors a and b is a vector a×b i j k b × a b =det ax ay az = α bx by bz a = i(aybz –azby)+ j(azbx –axbz)+ k(axby –aybx) ■ This vector is orthogonal to the plane formed by a and b (in the sense of a right-handed screw) ■ Its magnitude is the area of the parallelogram whose sides are a and b |a×b| = |a|·|b| |sinα| ■ For two vectors in the xy-plane the vector product can be used to determine convexity! How? Image formation Vectors: Mixed vector product ■ Mixed vector product of two vectors a c b and b is a scalar α ax ay az × a (a b)·c =det bx by bz cx cy cz ■ Its magnitude is the volume of the parallelepiped whose sides are a, b, and c Image formation Points and lines ■ We usually think of the equation of a line as specifying the set of points that lie on that line.
Load more

Recommended publications
  • Finite Projective Geometries 243
    FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig.
    [Show full text]
  • Natural Homogeneous Coordinates Edward J
    Advanced Review Natural homogeneous coordinates Edward J. Wegman∗ and Yasmin H. Said The natural homogeneous coordinate system is the analog of the Cartesian coordinate system for projective geometry. Roughly speaking a projective geometry adds an axiom that parallel lines meet at a point at infinity. This removes the impediment to line-point duality that is found in traditional Euclidean geometry. The natural homogeneous coordinate system is surprisingly useful in a number of applications including computer graphics and statistical data visualization. In this article, we describe the axioms of projective geometry, introduce the formalism of natural homogeneous coordinates, and illustrate their use with four applications. 2010 John Wiley & Sons, Inc. WIREs Comp Stat 2010 2 678–685 DOI: 10.1002/wics.122 Keywords: projective geometry; crosscap; perspective; parallel coordinates; Lorentz equations PROJECTIVE GEOMETRY Visualizing the projective plane is itself an intriguing exercise. One can imagine an ordinary atural homogeneous coordinates for projective Euclidean plane augmented by a set of points at geometry are the analog of Cartesian coordinates N infinity. Two parallel lines in Euclidean space would for ordinary Euclidean geometry. In two-dimensional meet at a point often called in elementary projective Euclidean geometry, we know that two points will geometry an ideal point. One could imagine that a always determine a line, but the dual statement, two pair of parallel lines would have an ideal point at each lines always determine a point, is not true in general end, i.e. one at −∞ and another at +∞. However, because parallel lines in two dimensions do not meet. there is only one ideal point for a set of parallel lines, The axiomatic framework for projective geometry not one at each end.
    [Show full text]
  • F. KLEIN in Leipzig ( *)
    “Ueber die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Coordinaten auf eine canonische Form,” Math. Ann 23 (1884), 539-578. On the transformation of the general second-degree equation in line coordinates into a canonical form. By F. KLEIN in Leipzig ( *) Translated by D. H. Delphenich ______ A line complex of degree n encompasses a triply-infinite number of straight lines that are distributed in space in such a manner that those straight lines that go through a fixed point define a cone of order n, or − what says the same thing − that those straight lines that lie in a fixed planes will envelope a curve of class n. Its analytic representation finds such a structure by way of the coordinates of a straight line in space that Pluecker introduced into science ( ** ). According to Pluecker , the straight line has six homogeneous coordinates that fulfill a second-degree condition equation. The straight line will be determined relative to a coordinate tetrahedron by means of it. A homogeneous equation of degree n between these coordinates will represent a complex of degree n. (*) In connection with the republication of some of my older papers in Bd. XXII of these Annals, I am once more publishing my Inaugural Dissertation (Bonn, 1868), a presentation by Lie and myself to the Berlin Academy on Dec. 1870 (see the Monatsberichte), and a note on third-order differential equations that I presented to the sächsischen Gesellschaft der Wissenschaften (last note, with a recently-added Appendix). The Mathematischen Annalen thus contain the totality of my publications up to now, with the single exception of a few that are appearing separately in the book trade, and such provisional publications that were superfluous to later research.
    [Show full text]
  • Homogeneous Representations of Points, Lines and Planes
    Chapter 5 Homogeneous Representations of Points, Lines and Planes 5.1 Homogeneous Vectors and Matrices ................................. 195 5.2 Homogeneous Representations of Points and Lines in 2D ............... 205 n 5.3 Homogeneous Representations in IP ................................ 209 5.4 Homogeneous Representations of 3D Lines ........................... 216 5.5 On Plücker Coordinates for Points, Lines and Planes .................. 221 5.6 The Principle of Duality ........................................... 229 5.7 Conics and Quadrics .............................................. 236 5.8 Normalizations of Homogeneous Vectors ............................. 241 5.9 Canonical Elements of Coordinate Systems ........................... 242 5.10 Exercises ........................................................ 245 This chapter motivates and introduces homogeneous coordinates for representing geo- metric entities. Their name is derived from the homogeneity of the equations they induce. Homogeneous coordinates represent geometric elements in a projective space, as inhomoge- neous coordinates represent geometric entities in Euclidean space. Throughout this book, we will use Cartesian coordinates: inhomogeneous in Euclidean spaces and homogeneous in projective spaces. A short course in the plane demonstrates the usefulness of homogeneous coordinates for constructions, transformations, estimation, and variance propagation. A characteristic feature of projective geometry is the symmetry of relationships between points and lines, called
    [Show full text]
  • Integrability, Normal Forms, and Magnetic Axis Coordinates
    INTEGRABILITY, NORMAL FORMS AND MAGNETIC AXIS COORDINATES J. W. BURBY1, N. DUIGNAN2, AND J. D. MEISS2 (1) Los Alamos National Laboratory, Los Alamos, NM 97545 USA (2) Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA Abstract. Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation consisting entirely of invariant submanifolds. A regular leaf, known as a flux surface,of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact, smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves including elliptic and hyperbolic magnetic axes. In this paper, the existence of exact, smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydro- dynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields; that they are presymplectic systems. Contents 1. Introduction 2 2. Summary of the Main Results 3 3. Field-Line Flow as a Presymplectic System 8 3.1. A perspective on the geometry of field-line flow 8 3.2. Presymplectic forms and Hamiltonian flows 10 3.3. Integrable presymplectic systems 13 arXiv:2103.02888v1 [math-ph] 4 Mar 2021 4.
    [Show full text]
  • Intro to Line Geom and Kinematics
    TEUBNER’S MATHEMATICAL GUIDES VOLUME 41 INTRODUCTION TO LINE GEOMETRY AND KINEMATICS BY ERNST AUGUST WEISS ASSOC. PROFESSOR AT THE RHENISH FRIEDRICH-WILHELM-UNIVERSITY IN BONN Translated by D. H. Delphenich 1935 LEIPZIG AND BERLIN PUBLISHED AND PRINTED BY B. G. TEUBNER Foreword According to Felix Klein , line geometry is the geometry of a quadratic manifold in a five-dimensional space. According to Eduard Study , kinematics – viz., the geometry whose spatial element is a motion – is the geometry of a quadratic manifold in a seven- dimensional space, and as such, a natural generalization of line geometry. The geometry of multidimensional spaces is then connected most closely with the geometry of three- dimensional spaces in two different ways. The present guide gives an introduction to line geometry and kinematics on the basis of that coupling. 2 In the treatment of linear complexes in R3, the line continuum is mapped to an M 4 in R5. In that subject, the linear manifolds of complexes are examined, along with the loci of points and planes that are linked to them that lead to their analytic representation, with the help of Weitzenböck’s complex symbolism. One application of the map gives Lie ’s line-sphere transformation. Metric (Euclidian and non-Euclidian) line geometry will be treated, up to the axis surfaces that will appear once more in ray geometry as chains. The conversion principle of ray geometry admits the derivation of a parametric representation of motions from Euler ’s rotation formulas, and thus exhibits the connection between line geometry and kinematics. The main theorem on motions and transfers will be derived by means of the elegant algebra of biquaternions.
    [Show full text]
  • The Intersection Conics of Six Straight Lines
    Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 46 (2005), No. 2, 435-446. The Intersection Conics of Six Straight Lines Hans-Peter Schr¨ocker Institute of Engineering Mathematics, Geometry and Informatics Innsbruck University e-mail: [email protected] Abstract. We investigate and visualize the manifold M of planes that intersect six straight lines of real projective three space in points of a conic section. It is dual to the apex-locus of the cones of second order that have six given tangents. In general M is algebraic of dimension two and class eight. It has 30 single and six double lines. We consider special cases, derive an algebraic equation of the manifold and give an efficient algorithm for the computation of solution planes. 1. Introduction Line geometry of projective three space is a well-established but still active field of geo- metric research. Right now the time seems to be right for tackling previously impossible computational problems of line space by merging profound theoretical knowledge with the computational power of modern computer algebra systems. An introduction and detailed overview of recent developments can be found in [5]. The present paper is a contribution to this area. It deals with conic sections that intersect six fixed straight lines of real projective three space P 3. The history of this problem dates back to the 19th century when A. Cayley and L. Cre- mona tried to determine ruled surfaces of degree four to six straight lines of a linear complex (compare the references in [4]).
    [Show full text]
  • 3 Homogeneous Coordinates
    3 Homogeneous coordinates Ich habe bei den folgenden Entwicklungen nur die Absicht gehabt [...] zu zeigen, dass die neue Methode [...] zum Beweise einzelner Stze und zur Darstellung allgemeiner Theorien sich sehr geschmeidig zeigt. Julius Pl¨ucker, Ueber ein neues Coordinatensystem, 1829 3.1 A spatial point of view Let K be any field1. And let K3 the vector space of dimension three over this field. We will prove that if we consider the one dimensional subspaces of K3 as points and the two dimensional subspaces as lines, then we obtain a projective plane by defining incidence as subspace containment. We will prove this fact by creating a more concrete coordinate representa- tion of the one- and two-dimensional subspaces of K3. This will allow us to be able to calculate with these objects easily. For this we first form equivalence classes of vectors by identifying all vectors v K3 that differ by a non-zero multiple: ∈ [v] := v! K3 v! = λ v for λ K 0 . { ∈ | · ∈ \{ }} K3\{(0,0,0)} The set of all such equivalence classes could be denoted K\{0} ; all non- zero vectors modulo scalar non-zero multiples. Replacing a vector by its equiv- alence class preserves many interesting structural properties. In particular, two 1 This is almost the only place in this book where we will refer to an arbitrary field K. All other considerations will be much more “down to earth and refer to specific fields” – mostly the real numbers R or the complex numbers C 54 3 Homogeneous coordinates vectors v1,v2 are orthogonal if their scalar product vanishes: v1,v2 = 0.
    [Show full text]
  • Arxiv:1802.05507V1 [Math.DG]
    SURFACES IN LAGUERRE GEOMETRY EMILIO MUSSO AND LORENZO NICOLODI Abstract. This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the au- thors, about three important classes of surfaces in Laguerre geometry, namely L-isothermic, L-minimal, and generalized L-minimal surfaces. The quadric model of Lie sphere geometry is adopted for Laguerre geometry and the method of moving frames is used throughout. As an example, the Cartan–K¨ahler theo- rem for exterior differential systems is applied to study the Cauchy problem for the Pfaffian differential system of L-minimal surfaces. This is an elaboration of the talks given by the authors at IMPAN, Warsaw, in September 2016. The objective was to illustrate, by the subject of Laguerre surface geometry, some of the topics presented in a series of lectures held at IMPAN by G. R. Jensen on Lie sphere geometry and by B. McKay on exterior differential systems. 1. Introduction Laguerre geometry is a classical sphere geometry that has its origins in the work of E. Laguerre in the mid 19th century and that had been extensively studied in the 1920s by Blaschke and Thomsen [6, 7]. The study of surfaces in Laguerre geometry is currently still an active area of research [2, 37, 38, 42, 43, 44, 47, 49, 53, 54, 55] and several classical topics in Laguerre geometry, such as Laguerre minimal surfaces and Laguerre isothermic surfaces and their transformation theory, have recently received much attention in the theory of integrable systems [46, 47, 57, 60, 62], in discrete differential geometry, and in the applications to geometric computing and architectural geometry [8, 9, 10, 58, 59, 61].
    [Show full text]
  • 12 Notation and List of Symbols
    12 Notation and List of Symbols Alphabetic ℵ0, ℵ1, ℵ2,... cardinality of infinite sets ,r,s,... lines are usually denoted by lowercase italics Latin letters A,B,C,... points are usually denoted by uppercase Latin letters π,ρ,σ,... planes and surfaces are usually denoted by lowercase Greek letters N positive integers; natural numbers R real numbers C complex numbers 5 real part of a complex number 6 imaginary part of a complex number R2 Euclidean 2-dimensional plane (or space) P2 projective 2-dimensional plane (or space) RN Euclidean N-dimensional space PN projective N-dimensional space Symbols General ∈ belongs to such that ∃ there exists ∀ for all ⊂ subset of ∪ set union ∩ set intersection A. Inselberg, Parallel Coordinates, DOI 10.1007/978-0-387-68628-8, © Springer Science+Business Media, LLC 2009 532 Notation and List of Symbols ∅ empty st (null set) 2A power set of A, the set of all subsets of A ip inflection point of a curve n principal normal vector to a space curve nˆ unit normal vector to a surface t vector tangent to a space curve b binormal vector τ torsion (measures “non-planarity” of a space curve) × cross product → mapping symbol A → B correspondence (also used for duality) ⇒ implies ⇔ if and only if <, > less than, greater than |a| absolute value of a |b| length of vector b norm, length L1 L1 norm L L norm, Euclidean distance 2 2 summation Jp set of integers modulo p with the operations + and × ≈ approximately equal = not equal parallel ⊥ perpendicular, orthogonal (,,) homogeneous coordinates — ordered triple representing
    [Show full text]
  • In Particular, Line and Sphere Complexes – with Applications to the Theory of Partial Differential Equations
    “Ueber Complexe, insbesondere Linien- und Kugel-complexe, mit Anwendung auf die Theorie partieller Differentialgleichungen,” Math. Ann. 5 (1872), 145-208. On complexes −−− in particular, line and sphere complexes – with applications to the theory of partial differential equations By Sophus Lie in CHRISTIANIA Translated by D. H. Delphenich _______ I. The rapid development of geometry in our century is, as is well-known, intimately linked with philosophical arguments about the essence of Cartesian geometry, arguments that were set down in their most general form by Plücker in his early papers. For anyone who proceeds in the spirit of Plücker’s work, the thought that one can employ every curve that depends on three parameters as a space element will convey nothing that is essentially new, so if no one, as far as I know, has pursued this idea then I believe that the reason for this is that no one has ascribed any practical utility to that fact. I was led to study the aforementioned theory when I discovered a remarkable transformation that represented a precise connection between lines of curvature and principal tangent curves, and it is my intention to summarize the results that I obtained in this way in the following treatise. In the first section, I concern myself with curve complexes – that is, manifolds that are composed of a three-fold infinitude of curves. All surfaces that are composed of a single infinitude of curves from a given complex satisfy a partial differential equation of second order that admits a partial differential equation of first order as its singular first integral.
    [Show full text]
  • Ray-Triangle and Ray-Quadrilateral Intersections in Homogeneous Coordinates
    Ray-Triangle and Ray-Quadrilateral Intersections in Homogeneous Coordinates Pat Hanrahan Geometry Computing Group Minnesota Supercomputer Center Minneapolis, MN 55415 [email protected] Experience shows that operation with homogeneous coordinates produces, at least with beginners, something like physical discomfirt. Felix Klein Recent articles in Ray Tracing News have discussed solutions to the problem of in- tersecting a ray with a triangle using the triangle’s barycentric coordinates. This article shows yet another way to think about the ray-triangle intersection problem. The idea is to think of the barycentric coordinates of the intersection point, not as the ratio of areas, but rather as the as ratios of the volume of tetrahedra. This technique can also be used to derive formulae for ray-quadrilateral intersection by using the fact that associated with every planar quadrilateral is a unique triangle. Finally, I would like to show the connection between these results and Plucker line coordinates. This insight leads to quite an efficient algorithm. Most people studying computer graphics learn about homogeneous coordinates and projective geometry because they allow perspective transformations to be expressed as linear transformations involving 4x4 matrices. When writing a ray tracer, however, most authors have duplicated Turner Whitted’s original approach where eye rays are directly formed in world space, obviating the need for a explicit viewing transformation. As a result the geometric calculations are usually expressed using familiar vector algebra in a Euclidean coordinate system. However, homogeneous coordinates were not invented to just express projective transformations; they have the advantage that many geometric calculations have elegant solutions when expressed in term of these coordinates.
    [Show full text]