We begin another week by looking at another Calculus topic, this time remaining the field of Calculus 3 (or Multivariable Calculus). We will be discussing the topic of Quadric Surfaces.
Quadric surfaces is just a way of saying 3-D shapes. In general, a Quadric Surface is any surface in the form:
$Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$
It is just your quadratic equation in $\mathcal{R}^2$. Instead, for Quadric Surfaces, we need a surface, which is just adding in this third dimension, thus arriving at the general equation from above. The problem is that these surfaces change so much depending on the constants $A ... J$. We will discuss the general forms we may find in the wild, many of which will be very familiar to you!
All The Forms!
1. Ellipsoid:
An ellipsoid, which is similar to the 2-D ellipse, is a shape similar to some pill type of shape. The general equation is in the form of: $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
2. Cone:
A cone is a shape that looks exactly like an ice-cream cone (coming directly from the name). It is as if you have a triangle and then smooth down the sides in 3-D till the base was circular and the sides were completely smooth. The general equation of a cone is in the form of: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$
3. Cylinder:
A cylinder is the shape of a soda can. You just take a circle in 2-D space and give it height. The general form of the equation of a cylinder is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
4. Hyperboloid of One Sheet and Two Sheets:
The shape of a hyperboloid of one sheet has the shape of what clay looks like when you are wanting to mold it into a vase. If it was two sheets instead, there would just be a distance between the two shape, like two eye contacts. The equations of the general form are very similar, where the general for the first sheet is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$
And for two sheets, the general form is: $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
5. Elliptic Paraboloid:
The shape is similar to piece of candy corn, where the cross sections of the shape are either ellipses or circles. The general form for the elliptic paraboloid is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$
6. Hyperbolic Paraboloid:
Our last one surface is the hyperbolic paraboloid. The shape resembles a saddle shape, like one you may use to ride a horse. The general form for the hyperbolic paraboloid is: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}$
Now that we have our footing, along with the tools needed, we can go into the goal for this week. We are wanting to take a quadric surface and intersect it with a plane (this intersection is also called the "Trace" of the surface!). For example, lets look at a general case. Take the surface of a cone, $f(x,y,z) = x^2 + y^2 -z^2 =0$, and intersect with the plane $g(x,y) = x+y = 0$. We would then find the intersection looks like the image below! In particular, the red part is showing the original cone, while the light blue shows up the part separated by the intersection. The interesting part is by looking at the different cross sections, whether being a circle or triangle, we could see what this shape would look like in 2-D space!
Our Problem:
For this, I wanted to take a shape whose intersection with a plane shows something different from just the standard equations. I arrived at the equations:
$f_1 = .5x^2 + .5y^2 = z$ and
$f_2 = -.5x^2 - .5y^2 + 5 = z$
These are equations for elliptic paraboloids, where part of $f_2$ is exactly the reflection of $f_1$, but we added that five to make these two intersect beyond just the origin.
Below here is a screenshot of the surfaces and their intersection! Below this, you will also find what the shape looks like when intersected with the plane $g(x,y) = x+y$.
Now that we see this intersection, we want to make some sense of it. This is where Openscad is very useful, because they have a built in tool that will get rid of the space we are not interested in. In particular, it makes the plane a bit thicker, so we can "look inside" where this plane cuts the two elliptic paraboloids. Below is what you would find at this intersection, thanks to Openscad:
In particular, I picked this example because of what we found inside of the surfaces here. We find faces which resemble different ovals, all of different sizes. Also, depending on how you look through the intersection, you can see the steepness of the surface as it extends while also seeing the sharpness of the cut off.
Quadric surfaces is just a way of saying 3-D shapes. In general, a Quadric Surface is any surface in the form:
$Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$
It is just your quadratic equation in $\mathcal{R}^2$. Instead, for Quadric Surfaces, we need a surface, which is just adding in this third dimension, thus arriving at the general equation from above. The problem is that these surfaces change so much depending on the constants $A ... J$. We will discuss the general forms we may find in the wild, many of which will be very familiar to you!
All The Forms!
1. Ellipsoid:
An ellipsoid, which is similar to the 2-D ellipse, is a shape similar to some pill type of shape. The general equation is in the form of: $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
2. Cone:
A cone is a shape that looks exactly like an ice-cream cone (coming directly from the name). It is as if you have a triangle and then smooth down the sides in 3-D till the base was circular and the sides were completely smooth. The general equation of a cone is in the form of: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$
3. Cylinder:
A cylinder is the shape of a soda can. You just take a circle in 2-D space and give it height. The general form of the equation of a cylinder is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
4. Hyperboloid of One Sheet and Two Sheets:
The shape of a hyperboloid of one sheet has the shape of what clay looks like when you are wanting to mold it into a vase. If it was two sheets instead, there would just be a distance between the two shape, like two eye contacts. The equations of the general form are very similar, where the general for the first sheet is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$
And for two sheets, the general form is: $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
5. Elliptic Paraboloid:
The shape is similar to piece of candy corn, where the cross sections of the shape are either ellipses or circles. The general form for the elliptic paraboloid is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z}{c}$
6. Hyperbolic Paraboloid:
Our last one surface is the hyperbolic paraboloid. The shape resembles a saddle shape, like one you may use to ride a horse. The general form for the hyperbolic paraboloid is: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = \frac{z}{c}$
Now that we have our footing, along with the tools needed, we can go into the goal for this week. We are wanting to take a quadric surface and intersect it with a plane (this intersection is also called the "Trace" of the surface!). For example, lets look at a general case. Take the surface of a cone, $f(x,y,z) = x^2 + y^2 -z^2 =0$, and intersect with the plane $g(x,y) = x+y = 0$. We would then find the intersection looks like the image below! In particular, the red part is showing the original cone, while the light blue shows up the part separated by the intersection. The interesting part is by looking at the different cross sections, whether being a circle or triangle, we could see what this shape would look like in 2-D space!
Our Problem:
For this, I wanted to take a shape whose intersection with a plane shows something different from just the standard equations. I arrived at the equations:
$f_1 = .5x^2 + .5y^2 = z$ and
$f_2 = -.5x^2 - .5y^2 + 5 = z$
These are equations for elliptic paraboloids, where part of $f_2$ is exactly the reflection of $f_1$, but we added that five to make these two intersect beyond just the origin.
Below here is a screenshot of the surfaces and their intersection! Below this, you will also find what the shape looks like when intersected with the plane $g(x,y) = x+y$.
Now that we see this intersection, we want to make some sense of it. This is where Openscad is very useful, because they have a built in tool that will get rid of the space we are not interested in. In particular, it makes the plane a bit thicker, so we can "look inside" where this plane cuts the two elliptic paraboloids. Below is what you would find at this intersection, thanks to Openscad:
In particular, I picked this example because of what we found inside of the surfaces here. We find faces which resemble different ovals, all of different sizes. Also, depending on how you look through the intersection, you can see the steepness of the surface as it extends while also seeing the sharpness of the cut off.
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