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Analytic Geometry in 3D
An Introduction to 3D Shapes


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Points
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  • In Cartesian Space:
    • A point in \(\mathbb{R}^{3}\) is an ordered triple of numbers (called coordinates)
  • In Homogeneous Space:
    • A point is specified using four coordinates
Points (cont.)
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  • From Cartesian to Homogeneous:
    • The Cartesian point \((x,y,z) \in \mathbb{R}^{3}\) becomes \((x,y,z,1)\)
    • The Cartesian point \((x,y) \in \mathbb{R}^{2}\) becomes \((x,y,0,1)\)
  • From Homogeneous to Cartesian:
    • The homogeneous point \((x,y,z,w)\) becomes the Cartesian point \((x/w,y/w,z/w)\) if \(w\) is nonzero
Points (cont.)
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  • Homogeneous Points without a Corresponding Cartesian Point:
    • When \(w = 0\) the point is often said to be "at infinity"
  • Limits:
    • Consider the sequence of points \((1,2,0,1.000)\), \((1,2,0,0.100)\),\((1,2,0,0.010)\), \((1,2,0,0.001)\),...
    • This sequence corresponds to the sequence of Cartesian points \((1,2,0)\), \((10,20,0)\),\((100,200,0)\), \((1000,2000,0)\),...
    • This is a sequence of points along the line \(y = 2x\)
Lines
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  • Implicit Form:
    • A point and a perpendicular is not specific enough
  • Visualization:
    • images/line3d-implicitform.gif
Lines (cont.)
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  • Parametric Form:
    • Given \(\bs{p} \in \mathbb{R}^{3}\) and \(\bs{v} \in \mathbb{R}^{3}\), the parametric form of a line is the set of points given by \(\bs{p} + \lambda \bs{v}\)
  • Using Two Points:
    • As in 2D, we can use \(\bs{v} = \bs{q} - \bs{p}\) for the line between \(\bs{p}\) and \(\bs{q}\)
Lines (cont)
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  • In 2D:
    • Two lines either intersect or are parallel
  • In 3D:
    • Can be skew (i.e., neither intersect nor are parallel)
    • Finding the itersection involves the solution of a linear system with two equations and three unknowns
Planes
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  • Definition:
    • Given a point \(\bs{p} \in \mathbb{R}^{3}\) and a direction vector \(\bs{n} \in \mathbb{R}^{3}\) through \(\bs{p}\), the set of points \(x \in \mathbb{R}^{3}\) satisfying \(\bs{n} \cdot (\bs{x} - \bs{p}) = 0\) is called the plane defined by \(\bs{p}\) and \(\bs{n}\)
  • Terminology:
    • This is called the implicit form of the plane
    • If \(|| \bs{n} || = 1\) then \(\bs{n}\) is called the normal to the plane
Planes (cont.)
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  • Parametric Form:
    • Given a point \(\bs{p} \in \mathbb{R}^{3}\) and two direction vectors \(\bs{v} \mbox{ and } \bs{w}\), the parametric form of a plane is given by \(\bs{p} + \lambda \bs{v} + \mu \bs{w}\)
  • Using Three Points:
    • Set \(\bs{v} = \bs{q} - \bs{p}\) and \(\bs{w} = \bs{r} - \bs{p}\)
Planar Shapes
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  • Triangles:
    • Are planar (i.e., lie in a single plane)
  • Other Shapes:
    • Shapes formed from four or more vertices may not be planar (e.g., consider a folded piece of paper)
Halfspaces
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  • Creation:
    • A plane creates two halfspaces
  • One Case:
    • \(\bs{n} \cdot (\bs{x} - \bs{p}) \gt 0\)
  • Another Case:
    • \(\bs{n} \cdot (\bs{x} - \bs{p}) \lt 0\)
Polyhedra
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  • Definition:
    • The intersection of a finite number of halfspaces
  • Examples:
    • Tetrahedron (4 equilateral triangle faces)
    • Pentahedron (5 faces i.e., the square pyramid and the triangular prism)
Spheres
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  • Definition:
    • The set of all points within a given radius, \(r\), of a point, \(\bs{c}\)
  • Implicit Form:
    • \(\{ \bs{p} \in \mathbb{R}^{3}: (\bs{p} - \bs{c}) \cdot (\bs{p} - \bs{c}) - r^2 \leq 0\}\)
There's Always More to Learn
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