Coordinate Systems – 3D, part 2

Once again, if you want to locate a point in 3-dimensional space, you need 3 numbers. In my last post, the 2-D Cartesian coordinate system (sometimes called the rectangular coordinate system) was extended to 3-D by adding another axis that is perpendicular to the other two axes. A point is then located using the coordinates (x, y, z), (1,−2,3) for example. Here are two more ways to locate a 3-D point that uses the rectangular system as a backdrop.

Spherical Coordinate System

If you remember in the 2-D scenario, polar coordinates used an angle (𝜃) from the positive x-axis and a distance (r) from the origin to determine the location of a point. And equations to represent a plot of points that satisfied the relationship between these coordinates had r‘s and 𝜃’s in them. In 3-D, the spherical coordinate system extends this method.

There are different conventions here but they all use two angles and a distance. The mathematical convention is shown below:


Here, a point is located by an angle from the positive x-axis, 𝜃, (like in polar coordinates), an angle from the positive z-axis, 𝜑, and a distance, r, from the origin. A point in this system has coordinates (r, 𝜃, 𝜑). As with polar coordinates, there are curves that are more easily expressed in spherical coordinates. For example, a sphere of radius 4 centred at the origin can be easily expressed in spherical coordinates as r = 4:

Or how about:


There are other conventions for spherical coordinates, one of which you are very familiar with. Locating a point on the earth is typically done with two numbers, longitude and latitude. Longitude is the angle a location is from the agreed reference meridian that runs through Greenwich England, and latitude is its angle from the equator. If the origin is at the earth’s centre with the x-axis going through the reference meridian (called the prime meridian) and the z-axis going through the north pole, longitude is our 𝜃, latitude is 90° − 𝜑, and r is always the radius of the earth.

Another convention is used to locate earth satellites using angles right ascension (similar to longitude) and declination (similar to latitude) from an agreed earth centred coordinate system where the axes are fixed and do not rotate with the earth.

There are other variations of this coordinate system; these are just a few.

Cylindrical Coordinate System

You can think of the cylindrical coordinate system as the 2D polar system with an added z coordinate:


Different letters/Greek symbols can be used, but they all represent the same system. If you look at the xy plane above, you see that this is just the polar coordinate system that was explained in a previous post. To add the third dimension, just move up z units to the desired point (r, 𝜃, z).

Cylindrical coordinates are useful in putting objects that are symmetrical with respect to the z-axis. For example, a cylinder of radius 4 can be easily described with the equation r = 4:

r = 4

Another example is a cone: z = r:

z = r

Switching between rectangular, spherical, and cylindrical coordinates is a useful tool in calculus. An equation expressed in one of these systems may be unsolvable but solvable in a different system.

In my next post, I’ll describe a sneaky way to locate a point in 2 or 3-D with one number: parametric equations.

Coordinate Systems – 3D, part 1

Since we live in a 3 dimensional world, many problems we encounter in fields such as science and engineering, as well as others, are modelled mathematically using 3 variables, hence, 3D.

The first coordinate system introduced to students to handle 3 variables is an extension of the 2D Cartesian coordinate system. If another number line is added to the 2D system that is 90° t0 the previous 2 axes, with the origin coinciding with the other two origins, you have the 3D system. The third axis is called the z-axis. So a point now needs 3 numbers to place it in 3D space: (x,y,z). Frequently, to draw a 3D grid on a 2D surface, the y and z axes are drawn in he plane of the surface and the x axis is drawn in perspective to show that it is perpendicular to the surface. So placing a point in a 3D Cartesian frame is an artistic challenge for me but drawing dashed lines parallel to the axes helps:

There are other orientations of the 3 axes when showing them in 2D, but this is a very common one.

As with the 2D Cartesian coordinate system, equations relating the variables x, y, and z can be plotted, showing all the values of x, y, and z that make the equation true.

In 2D, a general equation of a line is ax + by = c, where the a, b, and c are specific numbers. For example, the set of points that satisfy the equation 2x -3y = 7, plot as a straight line. By extension, in 3D, the general linear equation is ax + by + cz = d. Though this is called a linear equation, it plots as a plane in 3D:

The 3D version of a circle in 2D is a sphere. The generic equation of a sphere of radius r centred at the origin is x2 + y2+ x2 = r2:

Very interesting shapes can be made using 3D graphs. Here are a few:

\[z = \pm \sqrt{0.4^2-\left(2-\sqrt{x^2+y^2}\right)^2}\]
\[z=4 e^{-\frac{1}{4} y^2} \sin (2 x)\]

As with 2D, there are other ways of locating points in 3D. I will present some of these in my next post.

Coordinate Systems – 2D, part 2

In my last post, I talked about the Cartesian coordinate system where a point or a set of points can be located using the two numbers (x, y). There is another popular coordinate system that also locates a point in 2D space.

In the graph below, I have plotted the point (5, 3) in the Cartesian coordinate system we now know very well. I have added a line from the origin to that point and noted that the line makes an angle 𝜃 with the x-axis and that the length of the line is r. I’ve also added perpendicular lines from the point to the x and y axes to show that similar right triangles are formed:

From this graph, you can see that the right triangles have sides of lengths 5 and 3 units. From the Pythagorean Theorem,


And from trigonometry:


Why did I do this? Another way to locate that same point is to 1) define a line (also called a ray) from the origin that is 30.96° from the x-axis then, 2) go along that line 5.83 units and stop. That is your point. Welcome to polar coordinates.

This system of locating a point in 2D is called “polar” because the origin is a “pole” from which all the rays that you can define radiate from. In the polar coordinate system, you also need two numbers to locate a point: r and 𝜃. Conventionally, a point in polar coordinates is given in the order (r, 𝜃).

The variable r is a point’s distance from the origin. 𝜃 is the angle measured from the postive x-axis: anti-clockwise is + and clockwise is −. Because angles repeat every 360° or 2𝜋 radians, a particular (r, 𝜃) for a point is not unique. For example, (2, 25°) locates the same point as (2, 385°).

Graphing relations is usually done by plotting r as a function of 𝜃. Just as in Cartesian coordinates, the polar graph of an equation between r and 𝜃 is a picture of all the points whose (r, 𝜃) coordinates satisfy the equation. For example, the graph below are all points that satisfy r = 2cos(2𝜃):

Notice how a grid of concentric circles (possible r values) and rays (possible 𝜃 values) is super-imposed on the x and y axes. This is a polar graph grid.

There are Cartesian graphs that are more easily expressed and plotted in polar coordinates (and vice-versa). One glaring example is a circle. In the Cartesian frame, the equation of a circle, centred at the origin, is


where r is the radius. For a circle of radius 2, the above equation would have 4 on the right side and the graph would be a circle of radius 2 centred at the origin. In polar coordinates, the same graph would be r = 2. This is a picture of all points that are 2 units away from the origin:

In orbital dynamics, polar plots are most useful plotting a 2-body orbit. What is meant by “2-body” will be the subject of another post. The path of most orbits of satellites around the earth, are approximated by the ellipse. In Cartesian coordinates, the equation of an ellipse is:


The parameters a and b determine the size and orientation (long side vertical or horizontal) of the ellipse. For example,

The problem with this plot is that the geometric centre of the ellipse is at the origin. The path of an earth satellite is not the path followed in this plot if the earth is at the origin. The earth is at one of two special points associated with an ellipse called foci (singular focus). It is more useful in orbital dynamics if the ellipse were plotted in polar coordinates. The polar equation of an ellipse (actually any conic shape which includes circles, parabolas, and hyperbolas) is


where p and e are parameters that determine the size and the shape (circle, ellipse, parabola, or hyperbola) of the orbit. The parameter p is the y-intercept on a superimposed Cartesian frame and we will limit e to be strictly between 0 and 1 which makes the equation plot as an ellipse. This equation, by the way, is called the orbit equation because it accurately describes the shape of any orbit between two point masses without being perturbed by other masses. An example of an elliptical orbit around the earth with a satellite at a particular position is:

This polar plot is more useful to describe orbits because the earth is at the origin and it shows three of the parameters commonly used to describe a satellite’s position and orbit: p (called the semi-latus rectum), e (called the eccentricity), and 𝜃 (called the true anomaly).

Polar plots can generate shapes that would be unwieldy to generate in the Cartesian frame:

There are other less popular 2D coordinate systems like the parabolic coordinate system. Here is what parabolic graph paper looks like:

I personally do not want to go there.

Coordinate Systems – 2D, part 1

How do you locate a point on a two-dimensional (2D) surface. Since we are now in two dimensions, it will take a minimum of 2 numbers to locate a point. As in the case for 1D, the 2D surface used can be flat (which this post talks about) or curved: for example the surface of the Earth where the most common system to locate a point is the Geographic Coordinate System using latitude and longitude (again, two numbers to locate a point).

Cartesian Coordinate System

The coordinate system most used by students of mathematics is the Cartesian Coordinate System. This was invented (and named after) René Descartes in the 17th century. This system is used in 3D as well as higher dimensions, but this post is limited to 2D. As most people best learn and retain mathematical concepts visually, this system of plotting was, and still is, indispensable in algebra, calculus, geometry, trigonometry, and many more subjects. So what is the Cartesian Coordinate System?

If you take two 1D number lines, one horizontal and the other vertical so that they are at 90° to one another and that their origins intersect, voilà, you have a Cartesian Coordinate System:

The system above also has a superimposed grid so that we can more easily located a point.

Conventionally, the horizontal line is called the x-axis, and the vertical one the y-axis. Note the negative numbers are to the left and down. A point on a plane which has this system of location, is said to have coordinates (x, y). Note that x is always first. So a general point (x, y) will have a position such that it is x units left or right of the y-axis and y units above or below the x-axis. Here are some examples:

Analysing points and shapes plotted on a Cartesian coordinate system is called Coordinate Geometry. The lengths and midpoints of plotted lines with defined endpoints can be calculated. But the much more interesting use of a 2D coordinate system is plotting all the points that satisfy a relation between x and y values. This is called plotting an equation.

Suppose you have a relationship (equation) x2 + y2 = 4. What are the values of x and y that satisfy this equation? There are an infinite number of (x, y) pairs that will solve this equation. For example, (0, 2) solves this equation because 02 + 22 = 4. Even though there are infinite solutions, we can draw a picture of all the points that do solve the equation:

As you can see, the set of all points that solve this equation plots as a circle of radius 2. Plots of other equation can look quite strange:

But it is important to remember that the (x, y) coordinates of any point on the graph of a relation, makes the equation true when you substitute those values into it.

The Cartesian coordinate system is not the only way to locate a point in 2D. I will talk about another popular 2D coordinate sytstem in my next post.

Coordinate Systems – 1D

Many of the posts I have written, had plots of functions or relations between two variables, usually x and y. Most of teaching algebra and calculus relies on graphs to illustrate concepts. These graphs are plots of all the points that satisfy an algebraic relation between the two (or more) variables. Behind these plots is the coordinate system used. This series of posts explores the different coordinate systems commonly used in maths. Let’s first look at a one dimension (1D) coordinate system.

1D means that one number is needed to locate a point. The most used 1D coordinate system is the number line:

Number lines can be vertical or even curvy, for example, to show distance along a path. Usually though, the number line is a straight horizontal line. But they all have some things on common. First, they have to have a reference point: a point from which all other points obtain their position. This point here and in all coordinate systems is called the origin. And second, there is a scale: the distance between the tick marks that allow us to place a point. In the example above, the scale is 1 unit between tick marks. For example, if we want to plot the variable x = 5, the plot would be

There are an infinite number of points on this line: an infinite number of tick marks and an infinite number of points between each tick mark. What are the kinds of numbers that can be plotted?

Any number on the number line is called a real number. This is an actual mathematical term to distinguish these from other types of numbers used in maths such as imaginary numbers (despite the name, imaginary numbers have a real meaning in science and engineering). The set of real numbers is represented by the symbol ℝ. There are several subsets of real numbers.

The first set of numbers you learned as a child were the natural numbers. These are the counting numbers 1, 2, 3, … but do not include 0. This set of numbers is given the symbol ℕ.

Then you learned about 0 and negative integers. Integers are whole numbers (no decimals or fraction parts) and include the natural numbers, 0, and the negative integers. This set of numbers is given the symbol ℤ. Why not 𝕀? Because 𝕀 is the symbol for imaginary numbers which are not real numbers and 𝕀 is also sometimes used to refer to irrational numbers which I will talk about soon. Notice that ℕ is a subset of ℤ which is a subset of ℝ.

The next type of real numbers is the set of rational numbers. These are numbers that can be put into the form p/q where p and q are integers. Any integer is a rational number like 2 since 2 can be written as 2/1. Any decimal number with a repeating pattern of decimals (even if that is a repeating 0) is a rational number. As ℝ is already used for real numbers, this set of numbers is given the symbol ℚ. This stands for quotient as p/q is a quotient (a maths term for division). All of the previous sets of numbers are subsets of ℝ.

That leaves the set of irrational numbers: the numbers that cannot be put into the form p/q. Numbers like 𝜋 or √2 are irrational and symbols like these are the only way to represent the exact values. They cannot be exactly represented as a decimal number as their decimal parts never repeat. There is no common symbol for these but ℙ or 𝕀 are sometimes used. There are few occasions where only irrational numbers are required, but a more common notation would be ℝ\ℚ which means “all real numbers except rational numbers”. Here is a nice picture of how all these types of real numbers are related:

It’s the irrational and some of the rational numbers that lie between the tick marks. So 𝜋 would be approximately

Plotting single points on the number line is rather boring. But it can also be used to indicate intervals of numbers like all the numbers between −6 and 2. This is shown as −6 < x < 2 where the endpoints are not included or −6 ≤ x ≤ 2 if both endpoints are included or a combination. When plotting these, an open circle means that the endpoint is not included and a filled in circle means that it is included. So −6 < x ≤ 2 would plotted

There’s not much else we can do when using the 1D number line, but we have a lot more options when expanding to 2D: to be continued.

Transforming Quadratic (Parabolas) Graphs

Forms of Quadratic Equations

Quadratic equations come in several generic forms (or patterns) but they all have several things in common:

  1. The highest power of x (the independent variable) is 2 when all expressions are expanded in polynomial form.
  2. The other integer powers, 1 meaning just x, and 0 meaning a constant term, may or may not be present. But the squared term must be present.
  3. Other powers (negative integers, and non-integers), cannot be present.

The most common general form of a quadratic equation is \[y=ax^2 +bx+c,\] where the ab, and c are constants specific in a particular equation. For example, \[y=3x^2 -2x+7.\] Here a = 3, b = -2, and c = 7. There are other forms as well:\[\begin{array}{l}
y=a{\left(x-h\right)}^2 +k
\end{array},\]where the unspecified constants a, and k are specific to each form and are not the same numbers when converting between each of these forms. The most basic quadratic equation is \[y=x^2.\]Choosing various values of x, then squaring them to get the corresponding y values, and plotting these on a Cartesian coordinate grid, creates the following curve:


This shape is called a parabola. All quadratic equations have this shape when plotted but their position, orientation, and scale may be different. Each form of quadratic equations have their advantages. This lesson however, will concentrate on the form \[y=a{\left(x-h\right)}^2 +k.\]

The ‘a’ Factor

So we will be looking at the quadratic form \[y=a{\left(x-h\right)}^2 +k.\] This form is called the turning point form. Let’s start out simple and look at the effect of a alone by setting the h and k to zero. This leaves us with the equation \[y=ax^2.\] This coefficient in front of the x2 term scales and orients the parabola. If a is negative, all the y values are now negative. This flips (reflects) the parabola across the x-axis. If a is a large number, greater than 1, then the y values are larger for a given x than the y values in the basic y = x2 parabola. This has the effect of making the parabola sharper, that is, it is dilated along the x-axis. If a is a fraction between -1 and 1, then the y values increase more slowly. This has the effect of making the parabola flatter which is also a dilation along the x-axis. Below are several graphs of y = ax2 for various values of a. The basic parabola is shown (dashed curve) for comparison:


Notice how the negative sign flips the parabola across the x-axis.

The k Effect

Now let’s look at the equation \[y=ax^2 +k.\]I have just added a k to the previous equation form. If you add or subtract a constant number to an equation, it just raises or lowers the graph of the equation by k units. This is independent of the effect that a has on the curve. Below are examples for two choices of k using an a that was used above for comparison:


The h Reaction

Now so far we have scaled and inverted our parabola and moved it up or down. What about moving it right or left? That’s what the h does in the form \[y=a{\left(x-h\right)}^2 +k.\] We get to this form by replacing x with x – h. Whenever this is done in any equation, as well as the quadratic equation, this moves the curve to the right h units if h  is positive or to the left if h is negative. But be careful. There is a negative sign in the form y=a(xh)2+k, so in y=a(x-3)2+k, h = 3 so this parabola is moved 3 units to the right. Whereas,  a(x+3)2+k can be thought of as a(x-(-3))2+k, so h is -3 and this moves the parabola 3 units to the left. The effect of h on the graph of a parabola is independent of the effects of a or k.

Below are two examples of the effect of h using the last example above for comparison:


Functional Notation, Part 2

Last time we saw that we can replace y in an equation with f(x) when y is alone on the left side of an equation:

y = f(x) = 3x² – 5x + 1

The above is an example of the function definition. Once defined, you replace all the x‘s on the right side with whatever is in the brackets on the left side, even if it is not a number. For example,

f(2) = 3(2)² – 5(2) + 1 = 3
f(a) = 3a² – 5a + 1

Even if the thing in the brackets is another expression, for example, an expression that is used in calculus a lot is x + h:

f(x+h) = 3(x + h)² – 5(x + h) + 1

And you can even use another function of x inside the brackets of another function. Like x, the letter f is used in the first instance for a function, but if other functions need to be defined as well, other letters are used:

f(x) = 3x² – 5x + 1
g(x) = x² – 7
f[g(x)] = f(x² – 7) = 3(x² – 7)² – 5(x² – 7) + 1
g[f(x)] = g(3x² – 5x + 1) = (3x² – 5x + 1)² – 7

The domain of a function is all the valid values of x that can be used. Many times, the domain of a function (like f(x) and g(x) above) is just any real number. But there are functions where you cannot use just any number. For example, consider


There is one value of x you cannot use. That value is 2 because that will make the denominator 0, and as you know, this will bring the maths police to your door. So the domain of this function is all real numbers except for 0.

Now consider


Another illegal operation is taking the square root of a negative number. The requirement for this function is that x – 2 has to be 0 or greater. For this to be true, x must be greater than or equal to 2. The phrase ” greater than or equal to” can be replaced by the maths symbol ≥. So the domain of this function is x ≥ 2.

There are other reasons why the domain of a function is restricted, but the most common things to look for is dividing by 0 or taking the square root (or any even root) of a negative number.

Functional Notation, Part 1

Any maths beyond algebra relies on something called functional notation. I used this in one of my posts on Newton’s Laws but more needs to be said if you are to be comfortable with it.

We have looked at many equations to date. Most involved x and y. For example,

y = 3x² – 5x + 1

This will plot as a parabola on an xy coordinate system. The plot is a picture of all (x, y) pairs of numbers that, when substituted in the above equation, will result in a true statement. For example, (0, 1) would be a point on that parabola because 1 = 3(0)² – 5(0) + 1 = 1.

Most of the equations we have looked at have y on the left side and all the other things with x and stand-alone numbers are on the right side. In this form, it is easy to choose a number to replace x with, then do the maths with that number on the right side to find the corresponding y to make the equation true. For example, let’s choose “1” in place of x. The corresponding y will be

y = 3x² – 5x + 1 = 3(1)² – 5(1) + 1 = 3 – 5 + 1 = -1

So (1, -1) is also a point on this equation’s plot.

This is frequently done: choose a value for x, then replace x with that value and do the maths on the right side and find the corresponding y. Notice that we are free to choose a value for x, but once we do, the value for the corresponding y is fixed. For that reason, x is called the independent variable and y is called the dependent variable: y depends on the x we choose.

If y depends on the x we choose, then another way to say this is that “y is a function of x“. The new functional notation makes use of this by replacing y with f(x), read “function of x“. So the functional form of the equation above is

f(x) = 3x² – 5x + 1

This will plot exactly the same but we would replace the y label on the vertical axis with f(x):

So now it is easy to ask the question “What is the value of the function if x = 0″ by just replacing the “x” in f(x) with “0”, that is, f(0):

f(0) = 3(0)² – 5(0) + 1 = 1

So, f(0) = 1. We saw above that f(1) = -1. So now you will see the general coordinate as (x, f(x)) instead of (x, y). This is just a difference in notation – the plot stays the same.

There are some properties of functions and a few more definitions that will be explored next time.

System of Equations, Part 2

So last time I solved a system of two equations using the substitution method where the information from one equation is inserted into the other equation. This is the method of choice if it is easy to solve for one of the unknowns. However, the example that I used also lends itself well to the other method: Elimination.

The elimination method, like the substitution method, uses the two equations to generate one equation with one unknown which can be solved. The system I worked on last time was:

x + y = 108
xy = 38

The elimination method is simply adding the two equations together with the goal of eliminating one of the unknowns. So let’s add these two equations:

x + y = 108
xy = 38
2x = 146

Notice that I now have an equation with one unknown. Dividing both sides of this new equation by 2 gives:

2x/2 = 146/2 ⟹ x = 73

the same answer as before. Now use this value for x in one of the original equations. Let’s use the first one:

x + y = 108 ⟹ y = 108 – x = 108 – 73 = 35

So we get (thankfully) the same solution as before. But this example is rather contrived in that the y‘s were conveniently of opposite signs in the given equations. So let’s consider the following system:

2x + 3y = 51
3x + 2y = 49

Adding these two equations together will just give us another equation with two unknowns. But just like we do with single variable equations, we can modify one or both of these. Let me take the first equation, and multiply it by – 2. Why I am doing this will soon be revealed:

-2(2x + 3y) = 51(-2) ⟹ -4x -6y = -102

I will now multiply the second equation by 3:

3(3x + 2y) = 49(3) ⟹ 9x + 6y = 147

Now let’s repeat the system, replacing the equations with the new ones:

-4x -6y = -102
9x + 6y = 147

Notice that if I now add these equations, the y variable will disappear:

-4x -6y = -102
9x + 6y = 147
5x = 45 ⟹ x = 45/5 = 9

I will now substitute this partial solution into the original second equation:

3x + 2y = 49 ⟹ 3(9) + 2y = 49 ⟹ 27 + 2y = 49 ⟹ 2y = 49 – 27
⟹ 2y = 22 ⟹ y = 11

So x = 9 and y = 11 solves both of these equations.

Both methods, substitution and elimination, can either be used to solve a system of equations, but one method may be less work than the other.

System of Equations, Part 1

The equations that I have solved so far, have been equations with one unknown (variable). Suppose we have two unknowns in a problem and wish to solve for both. There is a rule in maths that you have to have the same number of equations as the number of unknowns if you are to solve for all unknowns. These equations need to be independent. what do I mean by that?

Suppose I need to find two numbers that add up to 108. In equation form, I want to find x and y such that x + y = 108. So I need another piece of information (equation) if I need a specific solution. Suppose you say “I can get another equation by just multiplying both sides of the given equation by 2: 2x + 2y = 216.” The problem here is that this second equation depends on the first equation. This is what I meant by “independent”. The second equation must be completely new information.

So let’s say that these two numbers must also have a difference of 38 : xy = 38. This is a completely new (independent) requirement. So we now have the two equations:

x + y = 108
xy = 38

So what two numbers satisfy these two requirements? There are two main methods for solving this: Substitution and Elimination. Actually, there is a third method which uses matrices, but I will explain what a matrix is and how to use it in a subsequent post. Let’s first talk about the substitution method.

The substitution method is basically inserting information from one of the equations into the other equation. Let’s solve the second equation for x. That is, use algebra to get x on one side and everything else on the other side. If I add y to both sides of the second equation, I get

xy + y = 38 + yx = 38 + y

Now put this definition of x into the first equation. That is, replace x in the first equation with what x is equal to from the second equation:

x + y = 108 ⟹ 38 + y + y = 108 ⟹ 38 + 2y =108

Notice that this new equation, which combines the information of the two equations into one, is a single equation with a single unknown. This can now be solved with the techniques we have seen before:

38 + 2y =108 ⟹ 2y = 108 – 38 = 70 ⟹ y = 70/2 = 35

So y is 35. You can now use this solution into any of the equations involving x and y, then solve for x. Let’s use x = 38 + y:

x = 38 + y = 38 + 35 = 73

So the two numbers are 35 and 73. They both add up to 108 and subtract to equal 38.

I’ll start the next post doing another example.