By Marina Cohen

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Demonstrate that if the coordinates of points are expressed by rational numbers (whole numbers and fractions), then a reflection in a line defined by rational coefficients a, b, c can be computed using integer arithmetic. 25. 11) that if tan θ = −a/b, then sin θ = a/ a2 + b2 cos θ = −b/ a2 + b2 1/2 . 1 the model of the front of a house was defined by instancing the picture element Square with vertices (0, 0), (1, 0), (1, 1), and (0, 1). 5 units in the x-direction, followed by a translation of 3 units in the x-direction and 1 unit in the ydirection.

The methods used to determine the line through two distinct points and the point of intersection of two lines both involve the cross product. This is due to the duality between points and lines in the plane which relates results about points and lines to a dual result about lines and points. For example, the property “points r1 , r2 , and r3 are collinear if and only if r1 · (r2 × r3 ) = 0” has the dual property “lines 1 , 2 , and 3 are concurrent if and only if 1 · ( 2 × 3 ) = 0”. Investigate further the property of duality [24, pp78–80].

The line is uniquely defined by the coefficients a, b, and c, or any non-zero multiple ra, rb, and rc of them. Therefore, it is natural to specify the line by the homogeneous line coordinates = (a, b, c) . Homogeneous Coordinates andTransformations of the Plane 39 It is also useful to consider to be a vector known as the line vector . Since any non-zero multiple of defines the same line, only the direction of is of importance. Let P(X, Y, W ) be a point on the line. 12) may be expressed as the dot product · P = aX + bY + cW = 0 .

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3-D Shapes by Marina Cohen


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