Home > Articles > Web Development > HTML/CSS

  • Print
  • + Share This
This chapter is from the book

1.11. A Brief Math Primer

To do anything interesting with Canvas, you need a good understanding of basic mathematics, especially working with algebraic equations, trigonometry, and vectors. It also helps, for more complex applications like video games, to be able to derive equations, given units of measure.

Feel free to skim this section if you’re comfortable with basic algebra and trigonometry and you can make your way to pixels/frame given pixels/second and milliseconds/frame. Otherwise, spending time in this section will prove fruitful throughout the rest of this book.

Let’s get started with solving algebraic equations and trigonometry, and then we’ll look at vectors and deriving equations from units of measure.

1.11.1. Solving Algebraic Equations

For any algebraic equation, such as (10x + 5) × 2 = 110, you can do the following, and the equation will still be true:

  • Add any real number to both sides
  • Subtract any real number from both sides
  • Multiply any real number by both sides
  • Divide both sides by any real number
  • Multiply or divide one or both sides by 1

For example, for (10x + 5) × 2 = 110, you can solve the equation by dividing both sides by 2, to get: 10x + 5 = 55; then you can subtract 5 from both sides to get: 10x = 50; and finally, you can solve for x by dividing both sides by 10: x = 5.

The last rule above may seem rather odd. Why would you want to multiply or divide one or both sides of an equation by 1? In Section 1.11.4, “Deriving Equations from Units of Measure,” on p. 62), where we derive equations from units of measure, we will find a good use for that simple rule.

1.11.2. Trigonometry

Even the simplest uses of Canvas require a rudimentary understanding of trigonometry; for example, in the next chapter you will see how to draw polygons, which requires an understanding of sine and cosine. Let’s begin with a short discussion of angles, followed by a look at right triangles.

1.11.2.1. Angles: Degrees and Radians

All the functions in the Canvas API that deal with angles require you to specify angles in radians. The same is true for the JavaScript functions Math.sin(), Math.cos(), and Math.tan(). Most people think of angles in terms of degrees, so you need to know how to convert from degrees to radians.

180 degrees is equal to π radians. To convert from degrees to radians, you can create an algebraic equation for that relationship, as shown in Equation 1.1.

Equation 1.1. Degrees and radians

01equ01.jpg

Solving Equation 1.1 for radians, and then degrees, results in Equations 1.2 and 1.3.

Equation 1.2. Degrees to radians

01equ02.jpg

Equation 1.3. Radians to degrees

01equ03.jpg

π is roughly equal to 3.14, so, for example, 45 degrees is equal to (3.14 / 180) × 45 radians, which works out to 0.7853.

1.11.2.2. Sine, Cosine, and Tangent

To make effective use of Canvas, you must have a basic understanding of sin, cos, and tan, so if you’re not already familiar with Figure 1.21, you should commit it to memory.

Figure 1.21

Figure 1.21. Sine, cosine, and tangent

You can also think of sine and cosine in terms of the X and Y coordinates of a circle, as illustrated in Figure 1.22.

Figure 1.22

Figure 1.22. Radius, x, and y

Given the radius of a circle and a counterclockwise angle from 0 degrees, you can calculate the corresponding X and Y coordinates on the circumference of the circle by multiplying the radius times the cosine of the angle, and multiplying the radius by the sine of the angle, respectively.

1.11.3. Vectors

The two-dimensional vectors that we use in this book encapsulate two values: direction and magnitude; they are used to express all sorts of physical characteristics, such as forces and motion.

In Chapter 8, “Collision Detection,” we make extensive use of vectors, so in this section we discuss the fundamentals of vector mathematics. If you’re not interested in implementing collision detection, you can safely skip this section.

Near the end of Chapter 8 we explore how to react to a collision between two polygons by bouncing one polygon off another, as illustrated in Figure 1.23.

Figure 1.23

Figure 1.23. Using vectors to bounce one polygon off another

In Figure 1.23, the top polygon is moving toward the bottom polygon, and the two polygons are about to collide. The top polygon’s incoming velocity and outgoing velocity are both modeled with vectors. The edge of the bottom polygon with which the top polygon is about to collide is also modeled as a vector, known as a edge vector.

Feel free to skip ahead to Chapter 8 if you can’t wait to find out how to calculate the outgoing velocity, given the incoming velocity and two points on the edge of the bottom polygon. If you’re not familiar with basic vector math, however, you might want to read through this section before moving to Chapter 8.

1.11.3.1. Vector Magnitude

Although two-dimensional vectors model two quantities—magnitude and direction—it’s often useful to calculate one or the other, given a vector. You can use the Pythagorean theorem, which you may recall from math class in school (or alternatively, from the movie the Wizard of Oz), to calculate a vector’s magnitude, as illustrated in Figure 1.24.

Figure 1.24

Figure 1.24. Calculating a vector’s magnitude

The Pythagorean theorem states that the hypotenuse of any right triangle is equal to the square root of the squares of the other two sides, which is a lot easier to understand if you look at Figure 1.24. The corresponding JavaScript looks like this:

var vectorMagnitude = Math.sqrt(Math.pow(vector.x, 2) +
                                Math.pow(vector.y, 2));

The preceding snippet of JavaScript shows how to calculate the magnitude of a vector referenced by a variable named vector.

Now that you know how to calculate a vector’s magnitude, let’s look at how you can calculate a vector’s other quantity, direction.

1.11.3.2. Unit Vectors

Vector math often requires what’s known as a unit vector. Unit vectors, which indicate direction only, are illustrated in Figure 1.25.

Figure 1.25

Figure 1.25. A unit vector

Unit vectors are so named because their magnitude is always 1 unit. To calculate a unit vector given a vector with an arbitrary magnitude, you need to strip away the magnitude, leaving behind only the direction. Here’s how you do that in JavaScript:

var vectorMagnitude = Math.sqrt(Math.pow(vector.x, 2) +
                                Math.pow(vector.y, 2)),
    unitVector = new Vector();

    unitVector.x = vector.x / vectorMagnitude;
    unitVector.y = vector.y / vectorMagnitude;

The preceding code listing, given a vector named vector, first calculates the magnitude of the vector as you saw in the preceding section. The code then creates a new vector—see Chapter 8 for a listing of a Vector object—and sets that unit vector’s X and Y values to the corresponding values of the original vector, divided by the vector’s magnitude.

Now that you’ve seen how to calculate the two components of any two-dimensional vector, let’s see how you combine vectors.

1.11.3.3. Adding and Subtracting Vectors

It’s often useful to add or subtract vectors. For example, if you have two forces acting on a body, you can sum two vectors representing those forces together to calculate a single force. Likewise, subtracting one positional vector from another yields the edge between the two vectors.

Figure 1.26 shows how to add vectors, given two vectors named A and B.

Figure 1.26

Figure 1.26. Adding vectors

Adding vectors is simple: You just add the components of the vector together, as shown in the following code listing:

var vectorSum = new Vector();

vectorSum.x = vectorOne.x + vectorTwo.x;
vectorSum.y = vectorOne.y + vectorTwo.y;

Subtracting vectors is also simple: you subtract the components of the vector, as shown in the following code listing:

var vectorSubtraction = new Vector();

vectorSubtraction.x = vectorOne.x - vectorTwo.x;
vectorSubtraction.y = vectorOne.y - vectorTwo.y;

Figure 1.27 shows how subtracting one vector from another yields a third vector whose direction is coincident with the edge between the two vectors. In Figure 1.27, the vectors A-B and B-A are parallel to each other and are also parallel to the edge vector between vectors A and B.

Figure 1.27

Figure 1.27. Subtracting vectors

Now that you know how to add and subtract vectors and, more importantly, what it means to do that, let’s take a look at one more vector quantity: the dot product.

1.11.3.4. The Dot Product of Two Vectors

To calculate the dot product of two vectors you multiply the components of each vector by each other, and sum the values. Here is how you calculate the dot product for two two-dimensional vectors:

var dotProduct = vectorOne.x * vectorTwo.x + vectorOne.y * vectorTwo.y;

Calculating the dot product between two vectors is easy; however, understanding what a dot product means is not so intuitive. First, notice that unlike the result of adding or subtracting two vectors, the dot product is not a vector—it’s what engineers refer to as a scalar, which means that it’s simply a number. To understand what that number means, study Figure 1.28.

Figure 1.28

Figure 1.28. A positive dot product

The dot product of the two vectors in Figure 1.28 is 528. The significance of that number, however, is not so much its magnitude but the fact that it’s greater than zero. That means that the two vectors point in roughly the same direction.

Now look at Figure 1.29, where the dot product of the two vectors is –528. Because that value is less than zero, we can surmise that the two vectors point in roughly different directions.

Figure 1.29

Figure 1.29. A negative dot product

The ability to determine whether or not two vectors point in roughly the same direction can be critical to how you react to collisions between objects. If a moving object collides with a stationary object and you want the moving object to bounce off the stationary object, you need to make sure that the moving object bounces away from the stationary object, and not toward the stationary object’s center. Using the dot product of two vectors, you can do exactly that, as you’ll see in Chapter 8.

That’s pretty much all you need to know about vectors to implement collision detection, so let’s move on to the last section in this brief math primer and see how to derive the equations from units of measure.

1.11.4. Deriving Equations from Units of Measure

As you will see in Chapter 5, motion in an animation should be time based, because the rate at which an object moves should not change with an animation’s frame rate. Time-based motion is especially important for multiplayer games; after all, you don’t want a game to progress more quickly for players with more powerful computers.

To implement time-based motion, we specify velocity in this book in terms of pixels per second. To calculate how many pixels to move an object for the current animation frame, therefore, we have two pieces of information: the object’s velocity in pixels per second, and the current frame rate of the animation in milliseconds per frame. What we need to calculate is the number of pixels per frame to move any given object. To do that, we must derive an equation that has pixels per frame on the left side of the equation, and pixels per second (the object’s velocity) and milliseconds per frame (the current frame rate) on the right of the equation, as shown in Equation 1.4.

Equation 1.4. Deriving an equation for time-based motion, part I

01equ04.jpg

In this inequality, X represents the animation’s frame rate in milliseconds/frame, and Y is the object’s velocity in pixels/second. As that inequality suggests, however, you cannot just multiply milliseconds/frame times pixels/second, because you end up with a nonsensical milliseconds-pixels/frame-seconds. So what do you do?

Recall the last rule we discussed in Section 1.11.1, “Solving Algebraic Equations,” on p. 54) for solving algebraic equations: You can multiply or divide one or both sides of an equation by 1. Because of that rule, and because one second is equal to 1000 ms, and therefore 1 second / 1000 ms is equal to 1, we can multiply the right side of the equation by that fraction, as shown in Equation 1.5.

Equation 1.5. Deriving an equation for time-based motion, part 2

01equ05.jpg

And now we are ready to move in for the kill because when you multiply two fractions together, a unit of measure in the numerator of one fraction cancels out the same unit of measure in the denominator of the other fraction. In our case, we cancel units of measure as shown in Equation 1.6.

Equation 1.6. Deriving an equation for time-based motion, part 3

01equ06.jpg

Canceling those units of measure results in Equation 1.7.

Equation 1.7. Deriving an equation for time-based motion, part 4

01equ07.jpg

Carrying out the multiplication results in the simplified equation, shown in Equation 1.8.

Equation 1.8. Deriving an equation for time-based motion, part 5

01equ08.jpg

Whenever you derive an equation, you should plug some simple numbers into your equation to see if the equation makes sense. In this case, if an object is moving at 100 pixels per second, and the frame rate is 500 ms per frame, you can easily figure out, without any equations at all, that the object should move 50 pixels in that 1/2 second.

Plugging those numbers into Equation 1.8 results in 500 × 100 / 1000, which equals 50, so it appears that we have a valid equation for any velocity and any frame rate.

In general, to derive an equation from variables with known units of measure, follow these steps:

  1. Start with an inequality, where the result is on the left, and the other variables are on the right.
  2. Given the units of measure on both sides of the equation, multiply the right side of the equation by one or more fractions, each equal to 1, whose units of measure cancel out the units of measure on the right side of the equation to yield the units of measure on the left side of the equation.
  3. Cancel out the units of measure on the right side of the equation.
  4. Multiply the fractions on the right side of the equation.
  5. Plug simple values whose result you can easily verify into the equation to make sure the equation yields the expected value.
  • + Share This
  • 🔖 Save To Your Account

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020