Inversion and Quartics

The task presented in this lesson is implemented in the environment of the Simplex software system.

A quartic is a fourth-order curve. For its unambiguous task requires fourteen points. These points are subject to restrictions. It is necessary that no three of them lie on one straight line, no six on the conic, no ten on the cube. A straight line crosses the quartic at a maximum of four points. As you can see, this is already a very difficult curve and, frankly, I have never seen its use in structural schemes. This, of course, does not mean that they don't know anything about it - fourth-order curves are used in aircraft construction, in which it is required to design surfaces with very smooth contours. But the calculation of such curves is mainly based on analytical dependencies, and not using constructive algorithms. In the Simplex, the quartic object is "embedded", although there is not even a third-order curve - cubes. As a result of some operations, for example, inversions, fourteen reference points of this curve can be obtained. Quartic names begin with the letter q. But while d Simplex there is not a single operation in which it could take part as an input parameter of the relation. And this happens only for the reason that its structural properties have not yet been investigated.

Today we will try to uncover some secrets of this curve, see how it manifests itself in interaction with other objects. But again, I repeat, these are only the most initial experiments and we will not make any global conclusions.

So where does the quartic come from?

Let's set a circle and a conic on the plane. The circle will be considered the circle of inversion. We place a set of points on the conic and transform them into inversions (Fig. 1). The blue line is a quartic.

Fig. 1. Getting a quartic by inverting a second-order curve with respect to a circle

An ellipse is a conic, a second-order curve. Inversion is a quadratic transformation. Multiplying the orders, we get the number four - the order of the result curve.

However, if you try now to cross the blue line with a straight line, then you are unlikely to find an option for its location at which you would get four points. The maximum that you can teach is two points, from which we can draw the erroneous conclusion that we have a second-order curve. By take your time! Just before us, the quartic has not yet appeared in all its glory.

Therefore, we move the inversion circle so that the curvature of the curve appears in greater completeness.

Fig. 2. The result of the inversion operation with the new position of the inversion circle

In the next figure, we see a curve formed by the inversion of the hyperbola (Fig. 3). Note that the quartic passes through the center of the inversion circle twice, since the hyperbola twice intersects the infinitely distant line.

Fig. 3. Quartic - the image of a hyperbola in inversion relative to a circle

In fig. 3, the place where the straight line crosses the quartic at four points is already very clearly visible. By the way, it is not at all difficult to determine these points. Draw a line and transform it in inverse with respect to the circle. As a result, we get a circle. Having found the points of intersection of the circle with the hyperbola, we transfer the result back (Fig. 4).

Fig. 4. Finding the points of intersection of a straight line with a quartic

Move the straight line below. Figure 5 shows the result of the intersection of a straight line with a quartic, in which we obtain two real and two complex conjugate points.

Рис. 5. Вариант пересечения, при котором образуются две вещественные и две комплексно сопряженные точки

Move the straight line below yet. In Fig. 6. we see four complex-valued points: two pairs of complex conjugate points.

Fig. 6. The intersection option, in which two pairs of complex conjugate points are formed

Choose a point on the conic. Draw a tangent to the conic at this point. Transforming the point and the line into inversion, we get a point on the quartic, as well as a circle tangent to the quartic at this point (Fig. 7). This problem can be solved in the "opposite" direction. We do not yet know how to choose a point on a quartic, but we can draw a line that we cross with a quartic and find the points of their intersection. Converting them to inversions, we get a point on the conic, draw a tangent to the conic, and return the result back.

Fig. 7. Drawing a circle tangent to the quartic at its point

Draw a tangent to the quartic at the conjugation point with the circle. Since the tangent of the quartic and the circle is common, we construct the tangent to the circle. Now it appears to us to build a normal to the quartic at its arbitrary point (Fig. 8).

Fig. 8. The construction of the tangent and the normal to the quartic at its point

Let us construct the own diameter of the hyperbola. Converting it to inversion, we get a circle. Since the inversion preserves angles, and the diameter passes at an angle of 90 degrees to the hyperbole, the transformed circle will also intersect the quartic at 90 degrees. Interestingly, the quartic, transformed relative to this circle, passes into itself (fig. 9).

Fig. 9. Demonstration of the existence of a circle orthogonal to the quartic

Now the basic principles of the study are explained. Now you can try to put your own experiments with the quartic.