Rotate backwards to stay level

It was a university programming assignment. I was to write an OpenGL program to render a Ferris wheel. The requirements were simple. There had to be 7 spokes emanating from the centre, each at an equal angle from each other. At the end of each spoke, there was to be a carriage. No outer rim was required. All 7 spokes and 7 carriages were to be simple cuboids. The wheel was to turn slowly. Colour aesthetics up to the individual.

I’ve already had lessons on simple rotation and translation operations in OpenGL. Ambient colouring, materials and shading were also taught. And simple cuboids were like basic rendering stuff.

The hard part that my fellow students found was in keeping the carriages level, while rotating the Ferris wheel.

My professor, being the evil mind that he was, chose 7 spokes, so the angle between each spoke was “weird” (no nice number). Believe it or not, that confused a heck of a lot of students… Rendering the spokes were easy. Render a long cuboid with one end at the origin, and rotate multiples of 360/7 degrees. The carriages on the other hand, needed some work…

A simple way of orienting the Ferris wheel is to align it with the XY plane, with the centre of the wheel at the origin. There are then 2 methods to render the carriages. The first is to calculate the XY coordinates of centres of all the carriages, and simply translate them there. Yes, there will be sines and cosines in the calculation. I’ll leave it to you as an exercise. If you were able to follow the article on bilinear interpolation in image rotation, you can do this.

The second method is to just use the rendering engine’s in-built functions. For example, you render a vertical spoke with one end at the origin, and the other end along the positive Y-axis. Then you render a carriage at the latter end of the spoke.

Then what do you do? Render the 2nd spoke-carriage combination exactly the same as the 1st, but rotate the whole thing 360/7 degrees clockwise. Here’s where the problem comes. Since the carriage is “tied” to the spoke, the rotation operation affects the carriage as well.

To keep the carriage level, you have to undo the rotation operation. How do you do that? Rotate the carriage in the other direction with the same angle.

Let’s leave the spokes out of the picture. To render a carriage in the correct position, at the correct angle, this is the series of steps to take (assuming the carriage is at the origin):

  • Rotate -i * (360/7) degrees (anti-clockwise)
  • Translate len units in positive Y direction
  • Rotate i * (360/7) degrees (clockwise)

where i is the number of multiples required, and len is the length of the spoke.

If you’re following this with OpenGL or DirectX, take care. That series of steps have to be reversed, because the 2 rendering engines apply the transformations in reverse order.

Hmm… that was a long story…

The one about chicken heads

Did you know chickens have this ability to keep their heads stable, even if their bodies are moving? Check this video out.

I believe chickens use a similar principle as discussed in the Ferris wheel. For example, if a chicken’s body was moved forwards (in the direction of its head), to keep its head in the same position, it has to move its head backwards.

A fluttering thought

To stay the same in the face of change, one must replicate and execute the change in the opposite manner.

Sort of like Newton’s First and Third Laws combined.

I was just thinking, in the face of changes in these times, staying the same is actually more tiring than going with the flow. I mean, you expend the exact same amount of effort to stay the same, and you have to expend more to improve (on business, on technology and so on). That doesn’t quite make sense…

It’s a cliche, I know, but the only constant in life is change. Expect it. Embrace it. Besides, staying the same is boring at some level…

A simple experiment

To convince yourself of the “backwards” principle discussed in the rendering of a Ferris wheel, try the following.

  • Stand up straight, face and body forwards
  • Turn your body, from the shoulders down, clockwise
  • You must keep your head still facing the same direction as at the start

Did you have to turn your head anti-clockwise to keep facing the same original direction?

Beginning C# – Loops and increments

Part of a computer’s charm is its ability to do repetitive tasks in a consistent manner. Doing lightning fast calculations? Good. Doing those calculations with accurate results? Better. Doing those calculations with accurate results everyday for the past year and not complaining? Oh my goodness!

The basis of programming this repetitive behaviour is the loop. There are 4 kinds of loops:

  • for loop
  • while loop
  • do-while loop
  • foreach loop

We’ll look at the for and while loops since they’re more common. In the following example code, we’ll be looking at the calculation of factorials. The factorial of 5, represented in mathematics as 5!, is equal to 1x2x3x4x5 = 120. From Wikipedia’s description of factorials,

In mathematics, the factorial of a non-negative integer n is the product of all positive integers less than or equal to n

Oh yes, the example code:

            const int cnFactorial = 7;
            // the following line of code will fail
            // because cnFactorial is a constant
            //cnFactorial = 5;

            int i, result;

            result = 1;
            for (i = 1; i <= cnFactorial; ++i)
            {
                result = result * i;
            }
            Console.WriteLine("Factorial for {0} is {1}", cnFactorial, result);

            result = 1;
            i = 1;
            while (i < cnFactorial + 1)
            {
                result = result * i;
                i++;
            }
            Console.WriteLine("Result is {1} for {0}!", cnFactorial, result);

            Console.WriteLine("End of program");
            Console.ReadLine();

Constants
Right there on the first line, we hit our new keyword const. It makes our integer variable cnFactorial a constant. Any value first assigned to cnFactorial can never be changed thereafter. Certain values are always the same, like PI (3.14159) or the number of months in a year (12). Adding the const keyword in front of a variable declaration fixes that value, reducing programming errors in case we super coders forget and happen to change an unchangeable value.

The resulting executable is also leaner because the computer compiler will, in effect, replace every occurrence of the constant variable’s name with the constant value. This means that program only has the constant value, and not the constant variable! Constant values take up less space in a program than variables.

Anatomy of a for loop
The for loop has 3 parts, separated by semicolons: initialiser, terminator and incrementor.
for (initialiser; terminator; incrementor)

The initialiser sets the starting conditions for the for loop. This is a good place to code any variable assignments such as resetting variable values.

The terminator is where you code the stopping condition for the for loop. Many a programmer has failed at least once, where they gave the wrong stopping condition and the for loop never ends. At best, the infinite loop keeps printing out messages. At worst, the computer crashes. Code with caution!

The incrementor is where something is usually changed, before checking the terminating condition.

It’s easier to just go through the code:

            result = 1;
            for (i = 1; i <= cnFactorial; ++i)
            {
                result = result * i;
            }

First, we assign the value of 1 to the variable “result”. Then in the initialiser of the for loop, we assign 1 to the variable i. There’s this unofficial type of variable which I’ll call the “loopers”. These looper variables are declared for the sole purpose of running through the for loop. They are also usually single characters by name. i, j, k and a, b, c and p, q, r and x, y, z are the more common ones.

Then the terminating condition reads “If i is less than or equal to cnFactorial, continue”.

The code in the incrementor, ++i, is a shortcut for “Add 1 to the value of i, then assign this new value to i”. It is equivalent to

i = i + 1;

Then the contents of the for loop reads “Multiply the value in variable ‘result’ by the value in variable ‘i’, and then assign this new value to ‘result’.”

The whole flow of the logic is like this:

  1. result has value of 1
  2. Start of for loop, i has value of 1
  3. Check that i (1) is less than or equal to cnFactorial (7), which is true so…
  4. Assign result (1) * i (1) which is 1 (1*1) to result
  5. Do ++i, so i now has value of 2
  6. Check that i (2) is less than or equal to cnFactorial (7), which is true so…
  7. Assign result (1) * i (2) which is 2 (1*2) to result
  8. Do ++i, so i now has value of 3
  9. Check that i (3) is less than or equal to cnFactorial (7), which is true so…
  10. Assign result (2) * i (3) which is 6 (2*3) to result
  11. Do ++i, so i now has value of 4
  12. Check that i (4) is less than or equal to cnFactorial (7), which is true so…
  13. Assign result (6) * i (4) which is 24 (6*4) to result
  14. Do ++i, so i now has value of 5
  15. Check that i (5) is less than or equal to cnFactorial (7), which is true so…
  16. Assign result (24) * i (5) which is 120 (24*5) to result
  17. Do ++i, so i now has value of 6
  18. Check that i (6) is less than or equal to cnFactorial (7), which is true so…
  19. Assign result (120) * i (6) which is 720 (120*6) to result
  20. Do ++i, so i now has value of 7
  21. Check that i (7) is less than or equal to cnFactorial (7), which is true so…
  22. Assign result (720) * i (7) which is 5040 (720*7) to result
  23. Do ++i, so i now has value of 8
  24. Check that i (8) is less than or equal to cnFactorial (7), which is false so we terminate the loop! Whoopee!

The purpose of that incredibly and mind-numbingly long and tedious explanation is so you can appreciate how obliging the computer is when performing repetitive tasks. Anyway, we then print the result. This time, we format the output a little.

            Console.WriteLine("Factorial for {0} is {1}", cnFactorial, result);

The {0} and {1} are placeholders, where {0} refers to the first value (cnFactorial) and {1} refers to the second value (result) after the string.

Why does {0} refer to the first value? Good question. Unfortunately I don’t have an easy answer. It is primarily due to the ancient programming custom of offsets, where the first item is offset by 0 positions. The second item is offset by 1 position. This has caused many beginner programmers to falter.

Homework
Unravel the logic for the while loop in the manner demonstrated above. Yes it’s tedious, but it’ll make you a better person. The while loop takes in only the terminating condition.

You’ll also notice the i++ and it does exactly the same thing as ++i. What’s the difference? Ahhh… but that is your second piece of homework. *grin*

Download source code.