On the Equi-projective Geometry of Paths

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Projective Geometry 2-D to 3-D

We add a unit vector in a third dimension, and view the Euclidean plane from the perspective of that point. For example each integer value on the x axis becomes an angle viewed from that viewpoint, as traced in blue lines below. The same principle works just as well in three dimensions, producing full 2-D perspective views of a 3-D scene. This principle makes perfect sense in perception, where an essentially infinite world projects an image onto a 2-D sensory surface.

The matrix encodes the entire projection including any translation, rotations, scaling, and shearing transformations that may be included in the projection. For example the coefficients along the main diagonal of the matrix, sx, sy, sz, determine the scaling of the x,y,z point in the projection, while the three values in the right column, dx, dy, dz, determine the translational displacement in three dimensions, etc.

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We can draw an analogy between the principle of projection and a laser beam mounted on gimbals that behaves like a Clifford vector located at the origin, that can point in any direction. If the laser beam is projected onto a screen, the linear beam is transformed into a point projected on the screen. Unlike the laser beam itself, the point is no longer confined to the origin, but can now be projected anywhere on the screen.

If we consider the projection itself, we see that the projection has transformed the basic elements of Clifford Algebra from linear vectors to punctate points. In projective geometry points are the most elemental feature. Translation of a in directions e1 and e2 is transformed to rotations of A in the rotational directions parallel to e1 and e2.

The projected origin P 0,0 is no longer a special point with the polarity reversals characteristic of Clifford vectors, the projected origin is now just another point in projective space. It is only the projection itself, P x,y or P x,y,z that represents a model in the brain, and in fact projection demonstrates how a finite representational medium, whether the surface of the retina or a volume in the visual cortex, can express a virtually infinite external world by depicting that world in projection.

Projected vectors in Geometric Algebra interact by the same sums and products as in Clifford Algebra although they seem qualitatively different in projection. If we allow the laser beam to modulate between the two beams, it creates a blend of the two points represented by a bivector surface between the two vectors, as by the wedge product between two vectors, and the projection of that bivector on the screen is the line joining the two points. The basic elements of projective geometry are points, and the wedge product between two points is the line segment that joins them.

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Like the points themselves, the bivector line segment can be presented at any location on the screen. Four or more vectors all projected onto the plane would fill in the polygonal area spanned by the points. The bivector between two points is anti-commutative, i. In the examples above for simplicity in visualization we have been dealing with projective geometry of a 2-D Euclidean space to a 3-D projective space and back to a 2-D projection.

While it is difficult to visualize a fourth spatial dimension and the projection process taking place within it, it is relatively easy to visualize the effect of that projection back in the 3-D projection. We will therefore ignore the details of the projection mechanism itself and study the properties of the projection back in the three dimensional world of our experience. This model world may look like a three-dimensional rectangular room, but it is supposed to represent a model that is distorted by perspective like an Ames Room , with objects in the distance presented at smaller scale than objects in the foreground.

The purpose of this projection is to project from an external world that is practically infinite in extent, into a model world of a finite size, small enough to fit in a human head. As with our earlier laser analogy, we can picture laser beams representing vectors projecting to points in three-dimensional space. Two points presented independently or successively represents the sum of those points in space.

On the Equi-projective Geometry of Paths.

The wedge product between two 3-D points defines the three-dimensional line segment that joins them. The wedge product between three or four points in the same plane defines a triangular, or rectangular plane in space. If we add another point outside of the plane, the wedge product between all those points defines the three-dimensional volume spanned by those points.

This is highly suggestive of the properties of our own world of perceptual experience. The operation of projection is very fundamental to Geometric Algebra. So much so that is is trivially easy to define a pinhole camera in Geometric Algebra.

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We begin with four points to define the image plane, whose surface is defined by the wedge product between them, i. Next we define a focal point F on one side of the image plane. The intersection between that line and the image plane can be computed simply as. The pinhole camera is complete! As an application, they draw and devise solutions to geometric division problems.

The notion of a mesh is not a standard one in projective geometry. Similarly, for this object we care about some points of intersection the four corners of the floor but not others the front left edge of the roof does not intersect the back right edge of the roof in a point in R3, even though their images intersect in Figure 2.

Squares in 2-point perspective. Instructor's manual for Chapter 7. Students discover the viewing circle; they locate the viewing target and viewing distance for a square in two-point perspectives. From there, they learn to draw a box and then a cube in 2-point perspective. We foreshadow the need for harmonic sets of points. Desargues Discovery. Instructor's manual for Chapter 8. Harmonic Homologies. We show that period-2 homologies like reflections relate to the above worksheet on squares in 2-point perspective.

Instructor's manual for Chapter 9. An introduction to cross ratio, with empirical evidence that it is a projective invariant. Harmonic sets of points and relationship to vanishing points of images of a rectangle.

Hermann Weyl

Applications to fence division and to photogrammetry. Instructor's manual for chapter Coordinate Geometry. Homogeneous Coordinates. Instructor's manual for Chapter The homogeneous coordinate worksheet introduces a more useful kind of coordinate system for perspective and projective images. The shape of Extended Real Space.

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Projective Geometry 5 Axioms, Duality and Projections

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On the Equi-projective Geometry of Paths On the Equi-projective Geometry of Paths
On the Equi-projective Geometry of Paths On the Equi-projective Geometry of Paths
On the Equi-projective Geometry of Paths On the Equi-projective Geometry of Paths
On the Equi-projective Geometry of Paths On the Equi-projective Geometry of Paths
On the Equi-projective Geometry of Paths On the Equi-projective Geometry of Paths
On the Equi-projective Geometry of Paths On the Equi-projective Geometry of Paths
On the Equi-projective Geometry of Paths On the Equi-projective Geometry of Paths

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