Introduction
Quadratic surfaces are 3-dimensional and can be seen on the xyz-plane. One way to think about this is that quadratic surfaces are the same as looking at a conic section in 3-dimentions. The equation to a quadratic surface is the following where A-J are constants \[Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0\] You may have noticed that each term doesn't exceed the exponent of 2. This is how we are able to tell that this is a quadratic surface. This equation is hard to understand but by rotations and translations the equation is modified, and we can get 7 possible outcomes.
Quadratic Surfaces
The possible outcomes are the following an ellipsoid, cylinder, cone, elliptic paraboloid, hyperboloid of 1 sheet, hyperboloid of 2 sheet, and the hyperbolic paraboloid. For each of these shapes they have an associated formula derived from the quadratic surface formula mentioned above.
Ellipsoid: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1\)
Cylinder: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=r\)
Cone: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\)
Elliptic Paraboloid: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=z\)
Hyperboloid of 1 sheet: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)
Hyperboloid of 2 sheet: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1\)
Hyperbolic Paraboloid: \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=z\)
From each of these formulae we can see that the exponent of each variable is a maximum of 2. To look at closely we will be using the hyperbolic paraboloid. In the image below we can see how a hyperbolic paraboloid can look where a = 2, b = 3.
Another thing to note is that the signs for each formula are interchangeable with each other. By doing this the shape does not change it only changes the orientation of how it lays on the xyz-plane.
Slicing A Quadratic Surface
We have looked at all the different shapes a quadratic surface can be. Now we will look at how an intersection to these surfaces would look like. Before we go into an example let's think about an ice-cream cone. We can get two shapes a circle and an ellipse depending on how we slice the cone. We can have different looking intersections on the same surface depending on how and where the plane goes through the surface. We will continue using the hyperbolic paraboloid.
In the image to the left our surface is split into two pieces by a plane. We know that the plane intersected our surface perpendicular to the vector (1,1,0). With this vector we are able to determine our plane
\begin{align*}
x*1 +y*1 +z*0 &=0
\\
x+y&=0
\\
y&=-x
\end{align*}
We have the equation of our plane which we can now use to find the equation of the intersection. In order to do this, we simply plug in the equation of the plane we found into our original formula for our surface. \begin{align*} z &= \frac{x^2}{4}-\frac{y^2}{9} \\ &=\frac{x^2}{4}-\frac{(-x)^2}{9} \\ z&=\frac{x^2}{4}-\frac{x^2}{9} = \frac{5x^2}{36} \end{align*} This is our equation to the intersection that was formed by the plane above. In the picture below we can clearly see how our intersection looks like a parabola. Which further shows our equation is correct which also is a parabola.
Why this Example
I chose particularly this quadratic surface because I was able to visualize this shape as a pringle. Being able to visualize the surface helps how the intersections can look like on this surface. The visualization helped me understand the different possible shape outcomes from this single quadratic surface. Choosing this surface also made it easier on finding the equation of the intersection because of how the z variable appeared in the formula.
Quadratic surfaces are 3-dimensional and can be seen on the xyz-plane. One way to think about this is that quadratic surfaces are the same as looking at a conic section in 3-dimentions. The equation to a quadratic surface is the following where A-J are constants \[Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0\] You may have noticed that each term doesn't exceed the exponent of 2. This is how we are able to tell that this is a quadratic surface. This equation is hard to understand but by rotations and translations the equation is modified, and we can get 7 possible outcomes.
Quadratic Surfaces
The possible outcomes are the following an ellipsoid, cylinder, cone, elliptic paraboloid, hyperboloid of 1 sheet, hyperboloid of 2 sheet, and the hyperbolic paraboloid. For each of these shapes they have an associated formula derived from the quadratic surface formula mentioned above.
Ellipsoid: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1\)
Cylinder: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=r\)
Cone: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\)
Elliptic Paraboloid: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=z\)
Hyperboloid of 1 sheet: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)
Hyperboloid of 2 sheet: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1\)
Hyperbolic Paraboloid: \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=z\)
From each of these formulae we can see that the exponent of each variable is a maximum of 2. To look at closely we will be using the hyperbolic paraboloid. In the image below we can see how a hyperbolic paraboloid can look where a = 2, b = 3.
Another thing to note is that the signs for each formula are interchangeable with each other. By doing this the shape does not change it only changes the orientation of how it lays on the xyz-plane.
Slicing A Quadratic Surface
We have looked at all the different shapes a quadratic surface can be. Now we will look at how an intersection to these surfaces would look like. Before we go into an example let's think about an ice-cream cone. We can get two shapes a circle and an ellipse depending on how we slice the cone. We can have different looking intersections on the same surface depending on how and where the plane goes through the surface. We will continue using the hyperbolic paraboloid.
We have the equation of our plane which we can now use to find the equation of the intersection. In order to do this, we simply plug in the equation of the plane we found into our original formula for our surface. \begin{align*} z &= \frac{x^2}{4}-\frac{y^2}{9} \\ &=\frac{x^2}{4}-\frac{(-x)^2}{9} \\ z&=\frac{x^2}{4}-\frac{x^2}{9} = \frac{5x^2}{36} \end{align*} This is our equation to the intersection that was formed by the plane above. In the picture below we can clearly see how our intersection looks like a parabola. Which further shows our equation is correct which also is a parabola.
Why this Example
I chose particularly this quadratic surface because I was able to visualize this shape as a pringle. Being able to visualize the surface helps how the intersections can look like on this surface. The visualization helped me understand the different possible shape outcomes from this single quadratic surface. Choosing this surface also made it easier on finding the equation of the intersection because of how the z variable appeared in the formula.
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