Pumpkin being cut

The quadric surface is derived from the quadratic function by adding an unknown number， We have all learned the quadratic function. It is just a two-dimensional plane transformed into a three-dimensional figure. If the quadric surface is intercepted by a parallel plane, its section is a quadratic curve. Usually, we call the surface represented by the quadratic equation of three variables as the quadric surface. The so-called three unknowns are the x, y, and z in the three-dimensional coordinate axis. Their powers are 1 or 2, which makes them form some fixed graphics, such as Ellipsoid, Hyperboloid of one sheet, Hyperbolic cylinder, and so on. In order to show it better, an ellipse closer to life would be a good example. For example, it’s almost Halloween and you decide to buy a pumpkin. If you find a great pumpkin, it can have an equation to show its appearance:

This equation is the coordinates of the circle of the plane plus the new unknown z, where a, b and c can be constants, So this equation is the equation of all ellipses, and different ellipses can be obtained by changing a, b, and c. You can also subtract or add some constants after x, y, z to change the position of the ellipse so that you can get the ellipses in different positions.

So in that case, for the sake of brevity, the position of the reunion will not be changed. We will bring in the constants, for example, a=b=4, c=3. Then our equation is obvious, then we can get a new equation and "pumpkin"

It can be seen that this is obviously not a ball, it should be flattened. Okay, now we have found a perfect "pumpkin". You bought it and decided to use it to make a pumpkin lantern after taking it back, so you have to open a flat mouth from above. If a=b, then all cuts parallel to the XY plane should be a circle because it does not depend on z at this time. oop, due to not being careful enough, the knife did not cut the pumpkin in parallel, but was crooked a lot, which resulted in the cutting plane not being parallel to the XY plane. Then suppose the equation of this knife is x+√3z=3, and this plane divides the previous model into two.

The incision is an ellipse, which is formed by the intersection of two equations. It can be seen that its tangent plane is also an ellipse. Then their intersection point is this ellipse, and the coordinates of the intersection point are:

This is one of the tangent planes, and for an ellipse with a=b (in the equation). Except for the tangent plane parallel to the XY plane, in other cases, all tangent planes should be an ellipse, all of which depend on the position of the plane.

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MA 391 Composition and Communication

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