Calculus/Conic sections

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Conic sections

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.

The three methods to intersect a cone with a plane. Note that the circle is a special form of an ellipse.

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.

Standard equationsEdit

EllipseEdit

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.

 
Ellipse: notations

DerivationEdit

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)

  (Simplify it)

 

  (Square both sides)

 

  (Simplify it)

  (Factor it)

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 propertiesEdit

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.

TransformationsEdit

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.

ParabolaEdit

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.

DerivationEdit

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 propertiesEdit

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  .

TransformationsEdit

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.

HyperbolaEdit

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.

DerivationEdit

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 propertiesEdit

 

  • 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

 

 

TransformationsEdit

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  

Axes rotationEdit

 
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.

 
An xy-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.

  1. The coordinate of the point is   in a   plane
  2. The distance between the point and the origin is  
  3. The angle between the line segment connecting the point and the origin and the positive   -axis is  
  4. The axes rotated   counterclockwise to establish a new plane:  
  5. 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 formEdit

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  ,  .

The rotated coordinates of the foci are

 

Now, we rotate the axes back to the original state.

 

 

 

Therefore, the foci are   and  .

 

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Conic sections