# Transformações 2D

Autor to artigo orginal em inglês: J David Eisenberg

Esta é uma tradução de 2D Transformations disponível em processing.org/tutorials mantendo a licença Creative Commons BY-NC-SA 4.0.

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Processing tem funções embutidas que tornam fácil você fazer objetos em um sketch mover, girar, ecrescer ou encolher. Este tutorial vai apresentar você às funções `translate`, `rotate`, e `scale`. De maneira que você as possa usar nos seus sketches.

As you know, your Processing window works like a piece of graph paper. When you want to draw something, you specify its coordinates on the graph.Here is a simple rectangle drawn with the code`rect(20, 20, 40, 40)`. The coordinatesystem (a fancy word for “graph paper”) is shown in gray. If you want to move the rectangle 60 units right and 80 units down,you can just change the coordinates by adding to the x and ystarting point:`rect(20 + 60, 20 + 80, 40, 40)` and the rectangle will appearin a different place. (We put the arrow in there for dramatic effect.) But there is a more interesting way to do it: move the graph paper instead.If you move the graph paper 60 units right and 80 units down,you will get exactly the same visual result. Movingthe coordinate system is called translation. The important thing to notice in the preceding diagram is that, as far asthe rectangle is concerned, it hasn’t moved at all. Its upper leftcorner is still at (20,20). When you use transformations, the things youdraw never change position; the coordinate system does.

Here is code that draws the rectangle in red by changing its coordinates,then draws it in blue by moving the grid. The rectangles are translucentso that you can see that they are (visually) at the same place. Only the method used tomove them has changed. Copy and paste this code into Processing and give it a try.

``````void setup()
{
size(200, 200);
background(255);
noStroke();

// draw the original position in gray
fill(192);
rect(20, 20, 40, 40);

// draw a translucent red rectangle by changing the coordinates
fill(255, 0, 0, 128);
rect(20 + 60, 20 + 80, 40, 40);

// draw a translucent blue rectangle by translating the grid
fill(0, 0, 255, 128);
pushMatrix();
translate(60, 80);
rect(20, 20, 40, 40);
popMatrix();
}

``````

Let’s look at the translation code in more detail.`pushMatrix()` is a built-in function thatsaves the current position of the coordinate system. The `translate(60, 80)` moves the coordinate system 60 units rightand 80 units down. The `rect(20, 20, 40, 40)` draws therectangle at the same place it was originally. Remember, the things you drawdon’t move—the grid moves instead. Finally,`popMatrix()` restores the coordinate system to the way it wasbefore you did the translate.

Yes, you could have done a `translate(-60, -80)` tomove the grid back to its original position. However, when you startdoing more sophisticated operations with the coordinate system, it’seasier to use `pushMatrix()` and `popMatrix()` tosave and restore the status rather than having to undo all your operations.Later on in this tutorial, you will find out why those functionsseem to have such strange names.

You may be thinking that picking up the coordinate system and moving itis a lot more trouble than just adding to coordinates. For a simple examplelike the rectangle, you are correct. But let’s take an example ofwhere `translate()` can make life easier. Here is somecode that draws a row of houses. It uses a loop that calls afunction named `house()`, which takesthe x and y location of the house’s upper-left corner as its parameters. ``````void setup()
{
size(400, 100);
background(255);
for (int i = 10; i < 350; i = i + 50)
{
house(i, 20);
}
}

``````

This is the code for drawing the house by changing itsposition. Look at all the additions that youhave to keep track of.

``````void house(int x, int y)
{
triangle(x + 15, y, x, y + 15, x + 30, y + 15);
rect(x, y + 15, 30, 30);
rect(x + 12, y + 30, 10, 15);
}

``````

Compare that to the version of the function that uses `translate()`.In this case, the code draws the house in the same place every time,with its upper left corner at (0, 0), and lets translation do all thework instead.

``````void house(int x, int y)
{
pushMatrix();
translate(x, y);
triangle(15, 0, 0, 15, 30, 15);
rect(0, 15, 30, 30);
rect(12, 30, 10, 15);
popMatrix();
}

``````

### Rotação

Além da translação que move a grade, é possível girar o sistema de coordenadas com a função `rotate()`. Essa função tem um parâmetro ou argumento, um número de radianos que você quer rodar. Em graus, um círculo tem 360°. Quando descrevemos os ângulos em radianos, a circuferência completa tem 2π radianos. Eis aqui um diagrama de como Processing mede ângulos em graus (preto) e radianos (vermelho). Since most people think in degrees, Processing has a built-in `radians()` functionwhich takes a number of degrees as its argument and converts it for you. It also hasa `degrees()` function that converts radians to degrees. Given that background,let’s try rotating a square clockwise 45 degrees.

``````void setup()
{
size(200, 200);
background(255);
smooth();
fill(192);
noStroke();
rect(40, 40, 40, 40);

pushMatrix();
fill(0);
rect(40, 40, 40, 40);
popMatrix();
}

``````

Hey, what happened? How come the square got moved and cut off?The answer is: the square did not move. The grid was rotated.Here is what really happened. As you can see, on the rotated coordinate system,the square still has its upper left corner at (40, 40). ### Girando da maneira certa

1. Faça a translação da origem do sistema de coordenadas (0, 0) para onde você quer que seja o canto superior esquerdo do quadrado. And here is the code and its result, without the grid marks.

``````void setup()
{
size(200, 200);
background(255);
smooth();
fill(192);
noStroke();
rect(40, 40, 40, 40);

pushMatrix();
// move the origin to the pivot point
translate(40, 40);

// then pivot the grid

// and draw the square at the origin
fill(0);
rect(0, 0, 40, 40);
popMatrix();
}

``````

And here is a program that generates a wheel of colors by usingrotation. The screenshot is reduced to save space.

``````void setup() {
size(200, 200);
background(255);
smooth();
noStroke();
}

void draw(){
if (frameCount % 10 == 0) {
fill(frameCount * 3 % 255, frameCount * 5 % 255,
frameCount * 7 % 255);
pushMatrix();
translate(100, 100);
rect(0, 0, 80, 20);
popMatrix();
}
}

``````

### Scaling

The final coordinate system transformation isscaling, which changes the size of the grid. Take a look at this example,which draws a square, then scales the grid to twice its normal size,and draws it again.

``````void setup()
{
size(200,200);
background(255);

stroke(128);
rect(20, 20, 40, 40);

stroke(0);
pushMatrix();
scale(2.0);
rect(20, 20, 40, 40);
popMatrix();
}

``````

First, you can see that the square appears to have moved. It hasn’t, ofcourse. Its upper left corner is still at (20, 20) on thescaled-up grid, but that point isnow twice as far away from the origin as it was in the original coordinatesystem. You can also see that the lines are thicker. That’s no opticalillusion—the lines really are twice as thick, because the coordinate systemhas been scaled to double its size.

Programming Challenge: Scale up the black square, but keep itsupper left corner in the same place as the gray square. Hint: use `translate()` to move the origin, then use `scale()`.

There is no law saying that you have to scale the x and y dimensionsequally. Try using `scale(3.0, 0.5)` to make the x dimensionthree times its normal size and the y dimension only half its normal size.

### Order Matters

When you do multiple transformations, the order makes a difference. A rotationfollowed by a translate followed by a scale will not give the same results as atranslate followed by a rotate by a scale. Here is some sample code andthe results.

``````void setup()
{
size(200, 200);
background(255);
smooth();
line(0, 0, 200, 0); // draw axes
line(0, 0, 0, 200);

pushMatrix();
fill(255, 0, 0); // red square
translate(70, 70);
scale(2.0);
rect(0, 0, 20, 20);
popMatrix();

pushMatrix();
fill(255); // white square
translate(70, 70);
scale(2.0);
rect(0, 0, 20, 20);
popMatrix();
}

``````

### The Transformation Matrix

Every time you do a rotation, translation, or scaling, the informationrequired to do the transformation is accumulated into a table ofnumbers. This table, or matrix has only a few rowsand columns, yet, through the miracle of mathematics, it contains all theinformation needed to do any series of transformations. And that’swhy the `pushMatrix()` and `popMatrix()` have thatword in their name.

### Push and Pop

What about the push and pop part of the names? These come from a computerconcept known as a stack, which works like a spring-loaded traydispenser in a cafeteria. When someone returns a tray to the stack, its weight pushes the platform down.When someone needs a tray, he takes it from the top of the stack,and the remaining trays pop up a little bit.

In a similar manner, `pushMatrix()` puts the current status ofthe coordinate system at the top of a memory area, and `popMatrix()`pulls that status back out. The preceding example used`pushMatrix()` and `popMatrix()` to make sure that thecoordinate system was “clean” before each part of the drawing.In all of the other examples, the calls to those two functions weren’treally necessary, but it doesn’t hurt anything to save and restorethe grid status.

Note: in Processing, the coordinate system is restored to its original state(origin at the upper left of the window, no rotation, and no scaling) everytime that the `draw()` function is executed.

### Three-dimensional Transforms

If you are working in three dimensions, you can call the`translate()` function with three arguments for thex, y, and z distances. Similarly, you cancall `scale()` with three arguments that tell howmuch you want the grid scaled in each of those dimensions.

For rotation, call the `rotateX()`, `rotateY()`, or `rotateZ()` functionto rotate around each of the axes. All three of these functionsexpect one argument: the number of radians to rotate.

### Case Study: An Arm-Waving Robot

Let’s use these transformations to animate a bluerobot waving its arms. Rather than try to write it all atonce, we will do the work in stages. The first step isto draw the robot without any animation.

The robot is modeled onthisdrawing, although it will not look as charming.First, we draw the robot so that itsleft and top side touch the x and y axes. Thatwill allow us to use `translate()` to easily placethe robot anywhere we want or to make multiple copies ofthe robot, as we did in the example of the houses.

When we refer to left and right in this drawing, we mean yourleft and right (the left and right side of your monitor),not the robot’s left and right.

``````void setup()
{
size(200, 200);
background(255);
smooth();
drawRobot();
}

void drawRobot()
{
noStroke();
fill(38, 38, 200);
rect(20, 0, 38, 30); // head
rect(14, 32, 50, 50); // body

rect(0, 32, 12, 37); // left arm
rect(66, 32, 12, 37); // right arm

rect(22, 84, 16, 50); // left leg
rect(40, 84, 16, 50); // right leg

fill(222, 222, 249);
ellipse(30, 12, 12, 12); // left eye
ellipse(47, 12, 12, 12); // right eye
}

`````` The next step is to identify the points wherethe arms pivot. That is shown in this drawing.The pivot points are (12, 32) and(66, 32). Note: the term “center of rotation”is a more formal term for the pivot point.

Now, separate the code for drawing the left and rightarms, and move the center of rotation for each arm to the origin, becauseyou always rotate around the (0, 0) point. To save space,we are not repeating the code for `setup()`.

``````void drawRobot()
{
noStroke();
fill(38, 38, 200);
rect(20, 0, 38, 30); // head
rect(14, 32, 50, 50); // body
drawLeftArm();
drawRightArm();
rect(22, 84, 16, 50); // left leg
rect(40, 84, 16, 50); // right leg

fill(222, 222, 249);
ellipse(30, 12, 12, 12); // left eye
ellipse(47, 12, 12, 12);  // right eye
}

void drawLeftArm()
{
pushMatrix();
translate(12, 32);
rect(-12, 0, 12, 37);
popMatrix();
}

void drawRightArm()
{
pushMatrix();
translate(66, 32);
rect(0, 0, 12, 37);
popMatrix();
}

``````

Now test to see if the arms rotate properly.Rather than attempt a full animation, we will justrotate the left side arm 135 degreesand the right side arm -45 degrees as a test. Hereis the code that needs to be added, and the result.The left side arm is cut off because of the windowboundaries, but we’ll fix that in the finalanimation.

``````void drawLeftArm()
{
pushMatrix();
translate(12, 32);
rect(-12, 0, 12, 37); // left arm
popMatrix();
}

void drawRightArm()
{
pushMatrix();
translate(66, 32);
rect(0, 0, 12, 37); // right arm
popMatrix();
}

``````

Now we complete the program by putting in the animation. Theleft arm has to rotate from 0° to 135° and back.Since the arm-waving is symmetric, theright-arm angle will always be the negative value ofthe left-arm angle. To make things simple,we will go in increments of 5degrees.

``````int armAngle = 0;
int angleChange = 5;
final int ANGLE_LIMIT = 135;

void setup()
{
size(200, 200);
smooth();
frameRate(30);
}

void draw()
{
background(255);
pushMatrix();
translate(50, 50); // place robot so arms are always on screen
drawRobot();
armAngle += angleChange;

// if the arm has moved past its limit,
// reverse direction and set within limits.
if (armAngle > ANGLE_LIMIT || armAngle < 0)
{
angleChange = -angleChange;
armAngle += angleChange;
}
popMatrix();
}

void drawRobot()
{
noStroke();
fill(38, 38, 200);
rect(20, 0, 38, 30); // head
rect(14, 32, 50, 50); // body
drawLeftArm();
drawRightArm();
rect(22, 84, 16, 50); // left leg
rect(40, 84, 16, 50); // right leg

fill(222, 222, 249);
ellipse(30, 12, 12, 12); // left eye
ellipse(47, 12, 12, 12);  // right eye
}

void drawLeftArm()
{
pushMatrix();
translate(12, 32);
rect(-12, 0, 12, 37); // left arm
popMatrix();
}

void drawRightArm()
{
pushMatrix();
translate(66, 32);
rect(0, 0, 12, 37); // right arm
popMatrix();
}

``````

### Case Study: Interactive Rotation

Instead of having the arms move on their own, we will modify the programso that the arms follow the mouse while the mouse button is pressed. Insteadof just writing the program at the keyboard, we first think about theproblem and figure out what the program needs to do.

Since the two arms move independently ofone another, we need to have one variable for each arm’s angle.It’s easy to figure out which arm to track. If the mouse is at theleft side of the robot’s center, track the left arm; otherwise,track the right arm.

The remaining problem is to figure out the angle of rotation. Given thepivot point position and the mouse position, how do you determine theangle of a line connecting those two points? The answer comes from the`atan2()` function, which gives (in radians) the angleof a line from the origin to a given y and x coordinate.In constrast to most other functions, the y coordinate comesfirst. `atan2()` returns a value from -π to π radians,which is the equivalent of -180° to 180°.

But what about finding the angle of a line that doesn’t startfrom the origin, such as the line from (10, 37) to (48, 59)?No problem; it’s the same as the angle of a line from(0, 0) to (48-10, 59-37). In general, to find theangle of the line from (x0, y0)to (x1, y1), calculate

``````   atan2(y1 - y0, x1 - x0)

``````

Because this is a new concept, rather than integrate it into the robotprogram, you should write a simple test program to see that you understand how `atan2()`works. This program draws a rectangle whose center of rotation isits upper left corner at (100, 100) and tracks the mouse.

``````void setup()
{
size(200, 200);
}

void draw()
{
float angle = atan2(mouseY - 100, mouseX - 100);

background(255);
pushMatrix();
translate(100, 100);
rotate(angle);
rect(0, 0, 50, 10);
popMatrix();
}

``````

That works great. What happens if we draw the rectangle so it istaller than it is wide? Change the preceding code to read`rect(0, 0, 10, 50)`. How come itdoesn’t seem to follow the mouse any more? The answer isthat the rectangle really is still following the mouse,but it’s the short side of the rectangle that does the following.Our eyes are trained to want the long side to be tracked. Becausethe long side is at a 90 degree angle to the short side,you have to subtract 90° (or π/2 radians) to get thedesired effect. Change the preceding code to read`rotate(angle - HALF_PI)` and try it again.Since Processing deals almost exclusively in radians,the language has defined the constants `PI` (180°),`HALF_PI` (90°), `QUARTER_PI` (45°)and `TWO_PI` (360°) for your convenience.

At this point, we can write the final version of thearm-tracking program. We start off with definitionsof constants and variables. The number 39in the definition of `MIDPOINT_X`comes from the fact that the body of the robot starts atx-coordinate 14 and is 50 pixels wide, so 39 (14 + 25)is the horizontal midpoint of the robot’s body.

``````/* Where upper left of robot appears on screen */
final int ROBOT_X = 50;
final int ROBOT_Y = 50;

/* The robot's midpoint and arm pivot points */
final int MIDPOINT_X = 39;
final int LEFT_PIVOT_X = 12;
final int RIGHT_PIVOT_X = 66;
final int PIVOT_Y = 32;

float leftArmAngle = 0.0;
float rightArmAngle = 0.0;

void setup()
{
size(200, 200);
smooth();
frameRate(30);
}

``````

The `draw()` function is next. It determines if themouse is pressed and the angle between the mouse locationand the pivot point, setting `leftArmAngle` and`rightArmAngle` accordingly.

``````void draw()
{
/*
* These variables are for mouseX and mouseY,
* adjusted to be relative to the robot's coordinate system
* instead of the window's coordinate system.
*/
float mX;
float mY;

background(255);
pushMatrix();
translate(ROBOT_X, ROBOT_Y); // place robot so arms are always on screen
if (mousePressed)
{
mX = mouseX - ROBOT_X;
mY = mouseY - ROBOT_Y;
if (mX < MIDPOINT_X) // left side of robot
{
leftArmAngle = atan2(mY - PIVOT_Y, mX - LEFT_PIVOT_X)
- HALF_PI;
}
else
{
rightArmAngle = atan2(mY - PIVOT_Y, mX - RIGHT_PIVOT_X)
- HALF_PI;
}
}
drawRobot();

popMatrix();
}

``````

The `drawRobot()` function remains unchanged, but aminor change to `drawLeftArm()` and `drawRightArm()`is now necessary. Because `leftArmAngle` and`rightArmAngle` are now computed in radians, thefunctions don’t have to do any conversion. The changesto the two functions are in bold.

``````void drawLeftArm()
{
pushMatrix();
translate(12, 32);
rotate(leftArmAngle);
rect(-12, 0, 12, 37); // left arm
popMatrix();
}

void drawRightArm()
{
pushMatrix();
translate(66, 32);
rotate(rightArmAngle);
rect(0, 0, 12, 37); // right arm
popMatrix();
}
``````