Find the equation of a conic passing through five point (using the Organic construction)

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Presentation transcript:

Find the equation of a conic passing through five point (using the Organic construction)

Example: Given five points: L1={(-4, -1), (-2, 3), (0, 2), (1, -5), (3, -2)}

Sort and Shift these points so that one will be on the origin: L2={(0, 0), (2, 4), (4, 3), (5, -4), 97, -1)}

Rotate all the points so that the last point will be on the x-axis: For example, if we want to move (7,-1) to the x-axis we must first must find the appropriate angle of rotation.

We use the following matrix to rotate the points such that one point will be moved to the x-axis.

Let us use this matrix to rotate (7,-1) to a point on the x-axis.

If we continue in the similar manner, we can find the rotation of all the points. Let the point be called

Let us find the parameters for the five point conic function. Points: A, B, C, D, and E. Angles:  =<CAB and  =<CBA

Directrix: We rotate line AE about A to create line AE’ and line BE about B to create AB’. Call the intersection of both of these line E’. In a similar manner create point D’. Line D’E’ will be our directrix for the organic construction.

Thus the directrix is the line passing through D’ and E’; (1)

Now, using  =<BAC and  =<ABC for our angles of rotation with the points of the directrix to get the following equation. ( 2 )

Graph of curve

The using the matrix of rotation we will rotate back the conic and the directrix.

Plug in x and y into (1), (2) and simplify the expression. (3) (4)

Translate the conic and directrix to the original position by using the following equalities.

Plug in x’ and y’ into (3), (4) and simplify the expression. (5) (6)