Projective Geometry
Projective Geometry
Coordinate projective geometry
Projective geometry is the study of geometric properties that are not changed by a projective transformation. A projective transformation is one that occurs when: points on one line are projected onto another line; points in a plane are projected onto another plane; or points in space are projected onto a plane, etc. Projections can be parallel or central.
For example, the sun shining behind a person projects his or her shadow onto the ground. Since the sun’s rays are for all practical purposes parallel, it is a parallel projection.
A slide projector projects a picture onto a screen. Since the rays of light pass through the slide, through the lens, and onto the screen, and since the lens acts like a point through which all the rays pass, it is a central projection. The lens is the center of the projection.
Some of the things that are not changed by a projection are collinearity, intersection, and order. If three points lie on a line in the slide, they will lie on a line on the screen. If two lines intersect on the slide, they will intersect on the screen. If one person is between two others on the slide, he or she will be between them on the screen.
Some of the things that are, or can be, changed by a projection are size and angles. One’s shadow is short in the middle of the day but very long toward sunrise and sunset. A pair of sticks that are crossed at right angles can cast shadows that are not at right angles.
Desargues’ theorem
Projective geometry began with Renaissance artists who wanted to portray a scene as someone actually on the scene might see it. A painting is a central projection of the points in the scene onto a canvas or wall, with the artist’s eye as the center of the projection (the fact that the rays are converging on the artist’s eye instead of emanating from it does not change the principles involved), but the scenes, usually Biblical, existed only in the artists’ imagination. The artists needed some principles of perspective to help them make their projections of these imagined scenes look real.
Among those who sought such principles was French engineer and mathematician Gerard Desargues (1591–1661). One of the many things he discovered was the remarkable theorem that now bears his name:
If two triangles ABC and A´B´C´ are perspective from a point (i.e., if the lines drawn through the corresponding vertices are concurrent at a point P), then the extensions of their corresponding sides will intersect in collinear points X, Y, and Z.
The converse of this theorem is also true: If two triangles are drawn so that the extensions of their corresponding sides intersect in three collinear points, then the lines drawn through the corresponding vertices will be concurrent.
It is not obvious what this theorem has to do with perspective drawing or with projections. If the two triangles were in separate planes, however, (in which case the theorem is not only true, it is easier to prove) one of the triangles could be a triangle on the ground and the other its projection on the artist’s canvas.
If, in Figure 1, BC and B´C´ were parallel, they would not intersect. If one imagines a point at infinity, however, they would intersect and the theorem would hold true. German astronomer and mathematician Johannes Kepler (1571–1630) is credited with introducing such an idea, but Desargues is credited with being the first to use it systematically. One of the characteristics of projective geometry is that two coplanar lines always intersect, but possibly at infinity.
Another characteristic of projective geometry is the principle of duality. It is this principle that connects Desargues’ theorem with its converse, although the connection is not obvious. It is more apparent in the three postulates that American mathematician Howard Whitley Eves (1911–2004) gives for projective geometry:
I. There is one and only one line on every two distinct points, and there is one and only one point on every two distinct lines.
II. There exist two points and two lines such that each of the points is on just one of the lines and each of the lines is on just one of the points.
III. There exist two points and two lines, the points not on the lines, such that the point on the two lines is on the line on the two points.
These postulates are not easy to read, and to really understand what they say, one should make drawings to illustrate them. Even without drawings, one can note that writing line in place of point and vice versa results in a postulate that says just what it said before. This is the principle of duality. One can also note that postulate I guarantees that every two lines will intersect, even lines that in Euclidean geometry would be parallel.
Coordinate projective geometry
If one starts with an ordinary Euclidean plane in which points are addressed with Cartesian coordinates, (x,y), this plane can be converted to a projective plane by adding a line at infinity. This is accomplished by means of homogeneous coordinates, (x1, x2, x3) where x = x1/x3 and y = x 2/x3. One can go back and forth between Cartesian coordinates and homogeneous coordinates quite easily. The point (7,3,5) becomes (1.4,0.6) and the point (4,1) becomes (4,1,1) or any multiple, such as (12,3,3) of (4,1,1).
One creates a point at infinity by making the third coordinate zero, for instance (4,1,0). One cannot convert this to Cartesian coordinates because (4/0,1/0) is meaningless. Nevertheless, it is a perfectly good projective point. It just happens to be “at infinity.” One can do the same thing with equations. In the Euclidean plane 3x - y + 4 = 0 is a line. Written with homogeneous coordinates 3x1/x3 -x2/x3 + 4 = 0 it is still a line. If one multiplies through by x3, the equation becomes 3x1 -x2 +4x3 = 0. The point (1,7) satisfied the original equation; the point (1,7,1) satisfies the homogeneous equation. So do (0,4) and (0,4,1) and so on.
In the Euclidean plane, the lines 3x - y + 4 = 0 and 3x - y + 10 = 0 are parallel and have no point in common. In homogeneous coordinates, they do. In homogeneous coordinates the system 3x1 -x2 +4x3 = 03x1 -x2 +10x3 = 0 does have a solution. It is (1,3,0) or any multiple of (1,3,0). Since the third coordinate is zero, however, this is a point at infinity. In the Euclidean plane, the lines are parallel and do not
intersect. In the projective plane, they intersect at infinity. The equation for the x-axis is y = 0; for the y-axis, it is x = 0. The equation for the line at infinity is correspondingly x3 = 0. One can use this equation to find where a curve crosses the line at infinity. Solving the system 3x1 - x2 + 4x3 = 0 x3 = 0 yields (1,3,0) or any multiple as a solution. Therefore, 3x1 - x2 + 4x3 = 0, or any line parallel to it, crosses at that point, as shown earlier.
Conic sections can be thought of as central projections of a circle. The vertex of the cone is the center of the projection and the generatrices of the cone are the rays along which the circle’s points are projected. One can ask where, if at all, the projection of a circle crosses the line at infinity.
A typical ellipse is x2 + 4y2 = 1. In homogeneous coordinates it is . Solving this with x3 =0yields , which has no solution other than (0,0,0), which is not a point in the projective plane.
A typical parabola is x2 - y = 0. In homogeneous coordinates this becomes x12 - x2x3 = 0. Solving this with x3 = 0 yields x1 = 0 and x2 = any number. The parabola intersects the line at infinity at the single point (0,1,0). In other words it is tangent to the line at infinity.
In a similar fashion, it can be shown that a hyperbola such as x2 - y2 = 1 crosses the line at infinity at two points, in this case (1,1,0) and (1,-1,0). These points, incidentally, are where the hyperbola’s asymptotes cross the line at infinity.
Cross ratio
Projections do not keep distances constant, nor do they enlarge or shrink them in an obvious way. in Figure 2, for instance, D´C´ is a little smaller than CD, but A´B´ is much larger than AB. There is, however, a rather obscure constancy about a projection’s effect on distance. It is known as the cross ratio. If A, B, C, and D are points in order on a line and if they are projected through a point P into points A´, B´, C´, and D´ on another line, then the two expressions and are equal.
Cross rations play an important part in many of projective geometry’s theorems.
Projective Geometry
Projective geometry
Projective geometry is the study of geometric properties which are not changed by a projective transformation. A projective transformation is one that occurs when: points on one line are projected onto another line; points in a plane are projected onto another plane; or points in space are projected onto a plane, etc. Projections can be parallel or central.
For example, the Sun shining behind a person projects his or her shadow onto the ground. Since the Sun's rays are for all practical purposes parallel, it is a parallel projection.
A slide projector projects a picture onto a screen. Since the rays of light pass through the slide, through the lens , and onto the screen, and since the lens acts like a point through which all the rays pass, it is a central projection. The lens is the center of the projection.
Some of the things that are not changed by a projection are collinearity, intersection, and order. If three points lie on a line in the slide, they will lie on a line on the screen. If two lines intersect on the slide, they will intersect on the screen. If one person is between two others on the slide, he or she will be between them on the screen.
Some of the things that are or can be changed by a projection are size and angles. One's shadow is short in the middle of the day but very long toward sunset. A pair of sticks which are crossed at right angles can cast shadows which are not at right angles.
Desargues' theorem
Projective geometry began with Renaissance artists who wanted to portray a scene as someone actually on the scene might see it. A painting is a central projection of the points in the scene onto a canvas or wall, with the artist's eye as the center of the projection (the fact that the rays are converging on the artist's eye instead of emanating from it does not change the principles involved), but the scenes, usually Biblical, existed only in the artists' imagination. The artists needed some principles of perspective to help them make their projections of these imagined scenes look real.
Among those who sought such principles was Gerard Desargues (1593-1662). One of the many things he discovered was the remarkable theorem which now bears his name:
If two triangles ABC and A'B'C' are perspective from a point (i.e., if the lines drawn through the corresponding vertices are concurrent at a point P), then the extensions of their corresponding sides will intersect in collinear points X, Y, and Z.
The converse of this theorem is also true: If two triangles are drawn so that the extensions of their corresponding sides intersect in three collinear points, then the lines drawn through the corresponding vertices will be concurrent.
It is not obvious what this theorem has to do with perspective drawing or with projections. If the two triangles were in separate planes, however, (in which case the theorem is not only true, it is easier to prove) one of the triangles could be a triangle on the ground and the other its projection on the artist's canvas.
If, in Figure 1, BC and B'C' were parallel, they would not intersect. If one imagines a "point at infinity," however, they would intersect and the theorem would hold true. Kepler is credited with introducing such an idea, but Desargues is credited with being the first to use it systematically. One of the characteristics of projective geometry is that two coplanar lines always intersect, but possibly at infinity .
Another characteristic of projective geometry is the principle of duality. It is this principle that connects Desargues' theorem with its converse, although the connection is not obvious. It is more apparent in the three postulates which Eves gives for projective geometry:
I. There is one and only one line on every two distinct points, and there is one and only one point on every two distinct lines.
II. There exist two points and two lines such that each of the points is on just one of the lines and each of the lines is on just one of the points.
III. There exist two points and two lines, the points not on the lines, such that the point on the two lines is on the line on the two points.
These postulates are not easy to read, and to really understand what they say, one should make drawings to illustrate them. Even without drawings, one can note that writing "line" in place of "point" and vice versa results in a postulate that says just what it said before. This is the principle of duality. One can also note that postulate I guarantees that every two lines will intersect, even lines which in Euclidean geometry would be parallel.
Coordinate projective geometry
If one starts with an ordinary Euclidean plane in which points are addressed with Cartesian coordinates, (x,y), this plane can be converted to a projective plane by adding a "line at infinity." This is accomplished by means of homogeneous coordinates, (x1,x2,x3) where x = x1/x3 and y = x 2/x3. One can go back and forth between Cartesian coordinates and homogeneous coordinates quite easily. The point (7,3,5) becomes (1.4,.6) and the point (4,1) becomes (4,1,1) or any multiple, such as (12,3,3) of (4,1,1).
One creates a point at infinity by making the third coordinate zero , for instance (4,1,0). One cannot convert this to Cartesian coordinates because (4/0,1/0) is meaningless. Nevertheless it is a perfectly good projective point. It just happens to be "at infinity." One can do the same thing with equations. In the Euclidean plane 3x - y + 4 = 0 is a line. Written with homogeneous coordinates 3x1/x3 - x2/x3 + 4 = 0 it is still a line. If one multiplies through by x3, the equation becomes 3x 1 - x2 + 4x3 = 0. The point (1,7) satisfied the original equation; the point (1,7,1) satisfies the homogeneous equation. So do (0,4) and (0,4,1) and so on.
In the Euclidean plane the lines 3x - y + 4 = 0 and 3x - y + 10 = 0 are parallel and have no point in common. In homogeneous coordinates they do. In homogeneous coordinates the system 3x1 - x2 + 4x3 = 0 3x1 - x2 + 10x3 = 0 does have a solution . It is (1,3,0) or any multiple of (1,3,0). Since the third coordinate is zero, however, this is a point at infinity. In the Euclidean plane the lines are parallel and do not intersect. In the projective plane they intersect "at infinity." The equation for the x-axis is y = 0; for the y-axis it is x = 0. The equation for the line at infinity is correspondingly x3 = 0. One can use this equation to find where a curve crosses the line at infinity. Solving the system 3x1 - x2 + 4x3 = 0 x3 = 0 yields (1,3,0) or any multiple as a solution. Therefore 3x1 - x 2 + 4x3 = 0, or any line parallel to it, crosses at that point, as we saw earlier.
Conic sections can be thought of as central projections of a circle . The vertex of the cone is the center of the projection and the generatrices of the cone are the rays along which the circle's points are projected. One can ask where, if at all, the projection of a circle crosses the line at infinity.
A typical ellipse is x2 + 4y 2 = 1. In homogeneous coordinates it is x 2 + 4x 211 = 0 yields x 2 + 4x 2 2 2 - x3 = 0. Solving this with x3 2 = 0, which has no solution other than (0,0,0), which is not a point in the projective plane.
A typical parabola is x2 - y = 0. In homogeneous coordinates this becomes x 21 - x2x3 = 0. Solving this with x3 = 0 yields x1 = 0 and x2 = any number. The parabola intersects the line at infinity at the single point (0,1,0). In other words it is tangent to the line at infinity.
In a similar fashion it can be shown that a hyperbola such as x 2 - y2 = 1 crosses the line at infinity at two points, in this case (1,1,0) and (1,-1,0). These points, incidentally, are where the hyperbola's asymptotes cross the line at infinity.
Cross ratio
Projections do not keep distances constant, nor do they enlarge or shrink them in an obvious way. In Figure 2, for instance, D'C' is a little smaller than CD, but A'B' is much larger than AB. There is, however, a rather obscure constancy about a projection's effect on distance . It is known as the "cross ratio." If A, B, C, and D are points in order on a line and if they are projected through a point P into points A', B', C', and D' on another line, then the two expressions and are equal.
Cross rations play an important part in many of projective geometry's theorems.
J. Paul Moulton