UnivHypGeom5: The circle and Cartesian coordinates
This video introduces basic facts about points, lines and the unit circle in terms of Cartesian coordinates. A point is an ordered pair of (rational) numbers, a line is a proportion (a:b:c) representing the equation ax+by=c, and the unit circle is x^2+y^2=1. With this notation we determine the line joining two points, the condition for colinearity of three points (using the determinant), the point where two non-parallel lines meet and the the condition for concurrency of three lines. We state the rational parametrization of the circle and show that a line meets a circle in either 1,2 or 0 points. These theorems are fundamental in applying Cartesian coordinates to Euclidean geometry and also, as we shall see, to hyperbolic geometry.CONTENT SUMMARY: Line through two? points theorem @04:31 Collinear points theorem @6:56 Determinants @08:09 Number system we will use @11:28 Concurrent lines theorem @15:10 Affinely parallel lines and Point on two lines theorem @16:42 Parameterization of unit circle theorem @19:33 experience with parameterization of unit circle @24:23 Meets of line and circle theorem @25:52 (THANKS to EmptySpaceEnterprise)
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