The three methods to intersect a cone with a plane. Note that the circle is a special form of an ellipse.
Conic sections are the intersections of a surface of a cone and a plane. There are three ways to intersect. The first method is to intersect the cone vertically, which the intersection will yield a hyperbola. The second method is to intersect the cone parallel to the outermost line of the cone, which will yield a parabola. The third method is to intersect the cone horizontally or slightly horizontally, which will yield an ellipse. For more information, see Conic section. If you have knowledge on this particular subject, you can help expand here.
In future chapters, you will encounter more about conic sections. As you progress into polar coordinates, parametric equations, and three-dimensional quadric surfaces, conic sections will again be a difficult subject to discuss. In this chapter, we will only talk about the basic Cartesian-coordinate conic sections.
Ellipses are shapes that have an interesting property. In order to find the standard equation for the ellipse, we must know what an ellipse is. Apart from the intersection of a inclined plane and the surface of a cone, there is another way to construct an ellipse.
Definition of an ellipse
An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.
Alternatively, assume points , , and , where is a constant. The ellipse is a set of points that satisfies:
The image on the right is the graph of an ellipse. If there is a point , then , where is a constant.
Knowing the defining characteristic of an ellipse, we can start find the equation.
To make things simple, let's set the center of the ellipse on the origin and the points and on the -axis. (see image on the right)
Since and , using the distance formula, we can get:
Now for the constant . Let the length of the semi-major axis be and the length of the semi-minor axis be . Imagine that point is now at the vertex, so . At this particular point,
Because the definition says that for any , we can safely say that . Thus, we can start solving the equation.
(Square both sides)
(Square both sides)
Finally, the equation is
This should be the equation. But can be further simplified. To do so, imagine again that our point is on the co-vertex, so . Thus,
Since we have already established that , we can write down an equation that links together:
Now we substitute as , we get the final result:
This equation is the standard form of the ellipse. It is considered standard because all key points are on the axes.
Terminology and properties
We already derived the equation. So, the terminology and properties will be based on that.
- Focus (plural: foci): the points which have coordinates respectively. The definition gives those points their function: .
- Semi-major axis: the axis with length .
- Semi-minor axis: the axis with length , .
- Vertex (plural: vertices): the endpoint of the semi-major axis. It has the coordinate .
- Co-vertex: the endpoint of the semi-minor axis. It has the coordinate , .
- Center: the middle point between the two foci. It has the coordinate .
Note that any changes towards the equation will change the coordinates for the key values. The coordinates above are strictly based upon the standard equation of the ellipse.
In the derivation, we stumbled upon a property, which is the relationship between the constants . The property is . In the derivation of the equation of the hyperbola, we will encounter this property again. However, it will be slightly different because of the signs. In the ellipse, , so the property ensures that . In the hyperbola, as we will see, , which means the length of the foci to the center is larger then that of the vertices to the center. In order to ensure that for convenience, the property will be slightly adjusted.
If we want the ellipse to be more "vertical" instead of "horizontal", the equation of the ellipse needs to be changed. To be more "vertical", the foci of the ellipse should be on the -axis, having coordinates . Using the same method for derivation, we get:
If we want the ellipse to translate (move without rotating) in the plane, using what we've learned in Chapter 1.2, we can modify the equation into:
, where is the center of the ellipse.
Parabolas can be interpreted as the more general form of the quadratic function. However, they are essentially different. While quadratic function is a function which describes a relationship between a variable and another, parabola is a curve in 2D space.
Definition of a parabola
Assume there is a point and a line that does not go through point . The parabola is a set of points that satisfies:
To make things simple for derivation, we put the point on the -axis and the vertex of the parabola on the origin (see image on the right), so Now, imagine that point is on the vertex, so . Because and , the equation for line is .
Now, we can start deriving the standard equation for the parabola.
Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.
Since we know that , , and , we can solve
So the equation can be turned into
(square both sides)
The standard equation for a parabola is
In Chapter 1.4, we talked about the various forms of quadratic functions, which if viewed as a geometric curve, is a parabola. To make the standard equation more familiar, we can adjust it to
Terminology and properties
Similar to the ellipse, we will use the standard equation to demonstrate key terms in the parabola.
- Focus: the point with coordinates .
- Directrix: the line with the equation .
- Vertex, the lowest point on the parabola with coordinates .
Recall that the vertex form of the quadratic function is . We can find that .
If we want the curve to face horizontally (the axis of symmetry is the -axis instead of the -axis), we change the coordinates for to and the line to . After some calculations, we get:
If we want to translate the parabola in the plane, using what we've learnt, we get:
, where the vertex is .
This resembles the vertex of the quadratic function . However, it is important to realize that parabolas and quadratic functions are fundamentally different. One is a geometric curve while the other is a transformation between two variables.
The relationship between the hyperbola and the reciprocal function is similar to that between the parabola and the quadratic function.
Definition of a hyperbola
Assume there are points and . A hyperbola is a set of points that satisfies:
Again, to make things easier, we put the points on the -axis and the center on the origin, so . Also, the length between the vertex and the center is . See the image on the right for more clarification.
The hyperbola with key values.
Since we know that , we can solve for .
Similar to the ellipse derivation case, we need to turn the constant into something in terms of either or . In this case, imagine point is on the vertex, so .
So, . Now, we derive.
Let's assume that , which means the curve is on the positive -axis, and
(square both sides)
(square both sides)
For convenience, we substitute as because looking at the graph, , and we want our constants to be positive. So
What about when ? That means the curve is on the negative -axis, and
Using the same method, we get:
Because we let , so
All possibilities have been discussed, and we can safely say that the equation of a hyperbola is
Terminology and properties
- Focus (plural: foci): the points .
- Vertex (plural: vertices): the endpoint of the semi-major axis. It has coordinates .
- Semi-major axis: the axis that has a length of .
- Asymptote: the line where the hyperbola is approaching but never intersects.
We briefly discussed the property of the hyperbola when talking ellipses. In that discussion, we said that we have to change the property slightly to make sure that all constants are positive. In this case, since , instead of the ellipse's , we have .
The asymptote is a little bit difficult to calculate because we need limits to find it.
Since , the relationship between and is:
Thus, the slope of the line passing through a point on the hyperbola and the center is:
The asymptote is the line where the hyperbola is approaching but never intersects. So, we can imagine that is infinitely large to a point that it intersects the asymptote (see Chapter 2.1 and 2.3 for more).
(where is infinitely large)
Formally, if we want to express "where is infinitely large", we write it like this: . We will discuss this expression in the next unit.
The slope of the asymptote is calculated to be .
As a result, the asymptote equation for the hyperbola is
If we want the hyperbola to face north-south instead of east-west, changing the coordinates of key points and using the same method for derivation will yield:
If we want to translate the hyperbola in the plane, the equation will become:
, where the center is and the asymptote is
The rotation of the x and y axes
When we start rotating curves like conic sections, it is extremely difficult to intuitively visualize the process of rotating curves: we are more used to translation than rotation. So, instead of rotating the curve back to something like , we rotate the axes to make the curve look like . Note that are the rotated coordinates.
-Cartesian coordinate system rotated through an angle
to an x'y'
-Cartesian coordinate system
In order to do so, we need to find out the relationship between the pre-rotation axes and the post-rotation axes. In other words, suppose there is a point with coordinates . After the rotation, the coordinates are . Express in terms of .
Now, we need to establish some coordinates and some of the properties.
- The coordinate of the point is in a plane
- The distance between the point and the origin is
- The angle between the line segment connecting the point and the origin and the positive -axis is
- The axes rotated counterclockwise to establish a new plane:
- Bullet points 3 and 4 can help us know that the angle between the line segment connecting the point and the origin and the positive -axis is
Now, we try to evaluate
According to trigonometry (see Chapter 1.3):
We can also evaluate in terms of :
Then substitute into the first two equations, we get:
Finally, the coordinate for the point after the rotation is .
The inverse transformation is .
General Cartesian form
The general Cartesian form for conic sections
- are constants
The best way to get an understanding of the general form is to look at the following example. It covers most of the content in this chapter and is relatively difficult.
Example: Find the foci for this conic section with the equation
To find the foci, we need the standard form. However, this is the general form. To make things worse, this is a rotated curve, which makes it impossible for humans to factor the equation into the translated standard form.
We know that the curve is rotated because after factoring, there is a factor. And the translated standard form does not have that particular factor. So, we should start rotating the axes so that after the rotation, the part can be cancelled.
First, we should list down what we know.
Then, we start rotating the axes.
Assume that we've rotated the axes into by rotating the axes by counterclockwise. The new rotated equation should be
Since after factoring the equation, we don't any factor, we need to make . To make it simpler, we just need to make .
Then, we need to express in terms of the constants , and so that we can know how much we should rotate the axes.
After substituting, we get:
After a lot of calculations and algebraic manipulations, we get:
We want .
Recall the trigonometric identity: double-angle formulae that
So, the equation can be turned into
For convenience, we will just solve for the simplest value.
Knowing the rotation angle and the other constants, we can solve for the new constants.
The new rotated equation for the curve is
We can now factor the equation into the translated standard form.
This is the standard equation for the ellipse. Recall that in an ellipse, .
Finally, we can determine the rotated coordinates of the foci.
The rotated coordinate of the center is
Because , so .
The rotated coordinates of the foci are
Now, we rotate the axes back to the original state.
Therefore, the foci are and .