| Popis: |
In geometric modeling of contours, in most cases it is convenient to use the equations of curves written in a parametric form. In this case, when modeling flat contours of the first order of smoothness, can be used circular arcs. The article proposes a method for determining the parametric equation of a circle (arc of a circle) for two ways: defining a circle over three points, as well as defining a circle over two points and a tangent in one of them. The definition of the parametric equations of circles in both cases was carried out using a projective coordinate system. In the first case, the conditions for the membership of three given points of this curve were imposed on the classical parametric equation of a second-order curve in the projective coordinate system to determine the unknown coefficients of the equation, and the given points were taken as three basis points of the projective system. Passing the obtained second-order curve through cyclic points made it possible to determine all unknown coefficients of the equation and, thus, determine the desired parametric equation of the circle in the projective coordinate system.In the second case, for simplicity, a local system of affine coordinates was chosen in which two given points were located on the Ox axis symmetrically with respect to the Oy axis, and the point on the tangent was located on the Oy axis. These 3 points were taken as 3 basis points of the projective coordinate system, and the fourth point of the projective coordinate system — the unit point — was defined in the affine system as a point belonging to this circle. The classical equation of a second-order curve in a projective coordinate system that passes through 3 basis points of the projective system and touches the coordinate lines in two of them, provided that the unit point belongs to the circle, is the desired equation of circle.After determining both of the desired equations in the projective coordinate system, using the transfer formulas from the projective system to the affine, the desired parametric equations of the circles in the affine system are determined. |
