DETERMINING NEW HIGHER ORDER CURVES USING BIQUADRATIC TRANSFORMATION METHODS

The article deals with fourth-order curves and their construction methods. For the first time, the concept of four-order curves is mentioned in the works of ancient Greek scientists. One such scientist was Perseus who lived in the second century. He obtained fourth-order curves by cutting a torus with planes parallel to the axis. And Nicomedes, a Greek geometer who lived in the third century, used fourth-order curves (Nicomedes shells) to solve segment problems. At the same time, he used fourth-order curves to solve trigonometric problems, so he called this curve the "Nicomedian envelope". Since the eighteenth century, scientists have tried to group fourth-order curves with different symbols. An earlier attempt to group fourth-order curves was made by the English mathematician Edward Waring. In 1694, fourth-order curves were also described in an article by the Swiss Daniel Bernoulli. He found a new fourth-order curve by equating the parameters a and c in this equation and studying the Cassini ellipse, which has two inflection points at the node. This curve was later called "Bernoulli's lemniscate". D.A. Gudkov and his students studied the shape of fourth-order curves, but the individual figure of fourth-order curves with separate material points and different geometric shapes. D.A. Gudkov obtained 24 types of the complete set of inseparable special forms of the fourth-order curve, whose singular points do not differ in this form, and obtained about 600 curves for these curves. At the beginning of the last century, Zeitana obtained 42 types of fourth- order curves without bending. In the article, the biquadratic transformation method was invented by the multivariate matching method, so we considered how to draw and obtain fourth-order curves using these biquadratic transformation methods. For this purpose, the method of some biquadratic transformations is used. Using some of these biquadratic transformations to obtain a fourth-order curve, the next case shows how the curve is constructed. In this article, we use the graphical model of the biquadratic transformation using the L8 method to obtain the fourth- order curve. To obtain the fourth-order curve, we first considered a straight line and then a circle as the initial image of the curve under different conditions.