Chapter 14The Document Object Model

Too bad! Same old story! Once you’ve finished building your house you notice you’ve accidentally learned something that you really should have known—before you started.

Friedrich Nietzsche, Beyond Good and Evil
Picture of a tree with letters and scripts hanging from its branches

When you open a web page in your browser, the browser retrieves the page’s HTML text and parses it, much like the way our parser from Chapter 12 parsed programs. The browser builds up a model of the document’s structure and uses this model to draw the page on the screen.

This representation of the document is one of the toys that a JavaScript program has available in its sandbox. It is a data structure that you can read or modify. It acts as a live data structure: when it’s modified, the page on the screen is updated to reflect the changes.

Document structure

You can imagine an HTML document as a nested set of boxes. Tags such as <body> and </body> enclose other tags, which in turn contain other tags or text. Here’s the example document from the previous chapter:

<!doctype html>
    <title>My home page</title>
    <h1>My home page</h1>
    <p>Hello, I am Marijn and this is my home page.</p>
    <p>I also wrote a book! Read it
      <a href="">here</a>.</p>

This page has the following structure:

HTML document as nested boxes

The data structure the browser uses to represent the document follows this shape. For each box, there is an object, which we can interact with to find out things such as what HTML tag it represents and which boxes and text it contains. This representation is called the Document Object Model, or DOM for short.

The global binding document gives us access to these objects. Its documentElement property refers to the object representing the <html> tag. Since every HTML document has a head and a body, it also has head and body properties, pointing at those elements.


Think back to the syntax trees from Chapter 12 for a moment. Their structures are strikingly similar to the structure of a browser’s document. Each node may refer to other nodes, children, which in turn may have their own children. This shape is typical of nested structures where elements can contain subelements that are similar to themselves.

We call a data structure a tree when it has a branching structure, has no cycles (a node may not contain itself, directly or indirectly), and has a single, well-defined root. In the case of the DOM, document.documentElement serves as the root.

Trees come up a lot in computer science. In addition to representing recursive structures such as HTML documents or programs, they are often used to maintain sorted sets of data because elements can usually be found or inserted more efficiently in a tree than in a flat array.

A typical tree has different kinds of nodes. The syntax tree for the Egg language had identifiers, values, and application nodes. Application nodes may have children, whereas identifiers and values are leaves, or nodes without children.

The same goes for the DOM. Nodes for elements, which represent HTML tags, determine the structure of the document. These can have child nodes. An example of such a node is document.body. Some of these children can be leaf nodes, such as pieces of text or comment nodes.

Each DOM node object has a nodeType property, which contains a code (number) that identifies the type of node. Elements have code 1, which is also defined as the constant property Node.ELEMENT_NODE. Text nodes, representing a section of text in the document, get code 3 (Node.TEXT_NODE). Comments have code 8 (Node.COMMENT_NODE).

Another way to visualize our document tree is as follows:

HTML document as a tree

The leaves are text nodes, and the arrows indicate parent-child relationships between nodes.

The standard

Using cryptic numeric codes to represent node types is not a very JavaScript-like thing to do. Later in this chapter, we’ll see that other parts of the DOM interface also feel cumbersome and alien. The reason for this is that the DOM wasn’t designed for just JavaScript. Rather, it tries to be a language-neutral interface that can be used in other systems as well—not just for HTML but also for XML, which is a generic data format with an HTML-like syntax.

This is unfortunate. Standards are often useful. But in this case, the advantage (cross-language consistency) isn’t all that compelling. Having an interface that is properly integrated with the language you are using will save you more time than having a familiar interface across languages.

As an example of this poor integration, consider the childNodes property that element nodes in the DOM have. This property holds an array-like object, with a length property and properties labeled by numbers to access the child nodes. But it is an instance of the NodeList type, not a real array, so it does not have methods such as slice and map.

Then there are issues that are simply poor design. For example, there is no way to create a new node and immediately add children or attributes to it. Instead, you have to first create it and then add the children and attributes one by one, using side effects. Code that interacts heavily with the DOM tends to get long, repetitive, and ugly.

But these flaws aren’t fatal. Since JavaScript allows us to create our own abstractions, it is possible to design improved ways to express the operations you are performing. Many libraries intended for browser programming come with such tools.

Moving through the tree

DOM nodes contain a wealth of links to other nearby nodes. The following diagram illustrates these:

Links between DOM nodes

Although the diagram shows only one link of each type, every node has a parentNode property that points to the node it is part of, if any. Likewise, every element node (node type 1) has a childNodes property that points to an array-like object holding its children.

In theory, you could move anywhere in the tree using just these parent and child links. But JavaScript also gives you access to a number of additional convenience links. The firstChild and lastChild properties point to the first and last child elements or have the value null for nodes without children. Similarly, previousSibling and nextSibling point to adjacent nodes, which are nodes with the same parent that appear immediately before or after the node itself. For a first child, previousSibling will be null, and for a last child, nextSibling will be null.

There’s also the children property, which is like childNodes but contains only element (type 1) children, not other types of child nodes. This can be useful when you aren’t interested in text nodes.

When dealing with a nested data structure like this one, recursive functions are often useful. The following function scans a document for text nodes containing a given string and returns true when it has found one:

function talksAbout(node, string) {
  if (node.nodeType == Node.ELEMENT_NODE) {
    for (let child of node.childNodes) {
      if (talksAbout(child, string)) {
        return true;
    return false;
  } else if (node.nodeType == Node.TEXT_NODE) {
    return node.nodeValue.indexOf(string) > -1;

console.log(talksAbout(document.body, "book"));
// → true

The nodeValue property of a text node holds the string of text that it represents.

Finding elements

Navigating these links among parents, children, and siblings is often useful. But if we want to find a specific node in the document, reaching it by starting at document.body and following a fixed path of properties is a bad idea. Doing so bakes assumptions into our program about the precise structure of the document—a structure you might want to change later. Another complicating factor is that text nodes are created even for the whitespace between nodes. The example document’s <body> tag does not have just three children (<h1> and two <p> elements) but actually has seven: those three, plus the spaces before, after, and between them.

So if we want to get the href attribute of the link in that document, we don’t want to say something like “Get the second child of the sixth child of the document body”. It’d be better if we could say “Get the first link in the document”. And we can.

let link = document.body.getElementsByTagName("a")[0];

All element nodes have a getElementsByTagName method, which collects all elements with the given tag name that are descendants (direct or indirect children) of that node and returns them as an array-like object.

To find a specific single node, you can give it an id attribute and use document.getElementById instead.

<p>My ostrich Gertrude:</p>
<p><img id="gertrude" src="img/ostrich.png"></p>

  let ostrich = document.getElementById("gertrude");

A third, similar method is getElementsByClassName, which, like getElementsByTagName, searches through the contents of an element node and retrieves all elements that have the given string in their class attribute.

Changing the document

Almost everything about the DOM data structure can be changed. The shape of the document tree can be modified by changing parent-child relationships. Nodes have a remove method to remove them from their current parent node. To add a child node to an element node, we can use appendChild, which puts it at the end of the list of children, or insertBefore, which inserts the node given as the first argument before the node given as the second argument.


  let paragraphs = document.body.getElementsByTagName("p");
  document.body.insertBefore(paragraphs[2], paragraphs[0]);

A node can exist in the document in only one place. Thus, inserting paragraph Three in front of paragraph One will first remove it from the end of the document and then insert it at the front, resulting in Three/One/Two. All operations that insert a node somewhere will, as a side effect, cause it to be removed from its current position (if it has one).

The replaceChild method is used to replace a child node with another one. It takes as arguments two nodes: a new node and the node to be replaced. The replaced node must be a child of the element the method is called on. Note that both replaceChild and insertBefore expect the new node as their first argument.

Creating nodes

Say we want to write a script that replaces all images (<img> tags) in the document with the text held in their alt attributes, which specifies an alternative textual representation of the image.

This involves not only removing the images but adding a new text node to replace them. Text nodes are created with the document.createTextNode method.

<p>The <img src="img/cat.png" alt="Cat"> in the
  <img src="img/hat.png" alt="Hat">.</p>

<p><button onclick="replaceImages()">Replace</button></p>

  function replaceImages() {
    let images = document.body.getElementsByTagName("img");
    for (let i = images.length - 1; i >= 0; i--) {
      let image = images[i];
      if (image.alt) {
        let text = document.createTextNode(image.alt);
        image.parentNode.replaceChild(text, image);

Given a string, createTextNode gives us a text node that we can insert into the document to make it show up on the screen.

The loop that goes over the images starts at the end of the list. This is necessary because the node list returned by a method like getElementsByTagName (or a property like childNodes) is live. That is, it is updated as the document changes. If we started from the front, removing the first image would cause the list to lose its first element so that the second time the loop repeats, where i is 1, it would stop because the length of the collection is now also 1.

If you want a solid collection of nodes, as opposed to a live one, you can convert the collection to a real array by calling Array.from.

let arrayish = {0: "one", 1: "two", length: 2};
let array = Array.from(arrayish);
console.log( => s.toUpperCase()));
// → ["ONE", "TWO"]

To create element nodes, you can use the document.createElement method. This method takes a tag name and returns a new empty node of the given type.

The following example defines a utility elt, which creates an element node and treats the rest of its arguments as children to that node. This function is then used to add an attribution to a quote.

<blockquote id="quote">
  No book can ever be finished. While working on it we learn
  just enough to find it immature the moment we turn away
  from it.

  function elt(type, ...children) {
    let node = document.createElement(type);
    for (let child of children) {
      if (typeof child != "string") node.appendChild(child);
      else node.appendChild(document.createTextNode(child));
    return node;

    elt("footer", "—",
        elt("strong", "Karl Popper"),
        ", preface to the second edition of ",
        elt("em", "The Open Society and Its Enemies"),
        ", 1950"));


Some element attributes, such as href for links, can be accessed through a property of the same name on the element’s DOM object. This is the case for most commonly used standard attributes.

But HTML allows you to set any attribute you want on nodes. This can be useful because it allows you to store extra information in a document. If you make up your own attribute names, though, such attributes will not be present as properties on the element’s node. Instead, you have to use the getAttribute and setAttribute methods to work with them.

<p data-classified="secret">The launch code is 00000000.</p>
<p data-classified="unclassified">I have two feet.</p>

  let paras = document.body.getElementsByTagName("p");
  for (let para of Array.from(paras)) {
    if (para.getAttribute("data-classified") == "secret") {

It is recommended to prefix the names of such made-up attributes with data- to ensure they do not conflict with any other attributes.

There is a commonly used attribute, class, which is a keyword in the JavaScript language. For historical reasons—some old JavaScript implementations could not handle property names that matched keywords—the property used to access this attribute is called className. You can also access it under its real name, "class", by using the getAttribute and setAttribute methods.


You may have noticed that different types of elements are laid out differently. Some, such as paragraphs (<p>) or headings (<h1>), take up the whole width of the document and are rendered on separate lines. These are called block elements. Others, such as links (<a>) or the <strong> element, are rendered on the same line with their surrounding text. Such elements are called inline elements.

For any given document, browsers are able to compute a layout, which gives each element a size and position based on its type and content. This layout is then used to actually draw the document.

The size and position of an element can be accessed from JavaScript. The offsetWidth and offsetHeight properties give you the space the element takes up in pixels. A pixel is the basic unit of measurement in the browser. It traditionally corresponds to the smallest dot that the screen can draw, but on modern displays, which can draw very small dots, that may no longer be the case, and a browser pixel may span multiple display dots.

Similarly, clientWidth and clientHeight give you the size of the space inside the element, ignoring border width.

<p style="border: 3px solid red">
  I'm boxed in

  let para = document.body.getElementsByTagName("p")[0];
  console.log("clientHeight:", para.clientHeight);
  console.log("offsetHeight:", para.offsetHeight);

The most effective way to find the precise position of an element on the screen is the getBoundingClientRect method. It returns an object with top, bottom, left, and right properties, indicating the pixel positions of the sides of the element relative to the top left of the screen. If you want them relative to the whole document, you must add the current scroll position, which you can find in the pageXOffset and pageYOffset bindings.

Laying out a document can be quite a lot of work. In the interest of speed, browser engines do not immediately re-layout a document every time you change it but wait as long as they can. When a JavaScript program that changed the document finishes running, the browser will have to compute a new layout to draw the changed document to the screen. When a program asks for the position or size of something by reading properties such as offsetHeight or calling getBoundingClientRect, providing correct information also requires computing a layout.

A program that repeatedly alternates between reading DOM layout information and changing the DOM forces a lot of layout computations to happen and will consequently run very slowly. The following code is an example of this. It contains two different programs that build up a line of X characters 2,000 pixels wide and measures the time each one takes.

<p><span id="one"></span></p>
<p><span id="two"></span></p>

  function time(name, action) {
    let start =; // Current time in milliseconds
    console.log(name, "took", - start, "ms");

  time("naive", () => {
    let target = document.getElementById("one");
    while (target.offsetWidth < 2000) {
  // → naive took 32 ms

  time("clever", function() {
    let target = document.getElementById("two");
    let total = Math.ceil(2000 / (target.offsetWidth / 5));
    target.firstChild.nodeValue = "X".repeat(total);
  // → clever took 1 ms


We have seen that different HTML elements are drawn differently. Some are displayed as blocks, others inline. Some add styling—<strong> makes its content bold, and <a> makes it blue and underlines it.

The way an <img> tag shows an image or an <a> tag causes a link to be followed when it is clicked is strongly tied to the element type. But we can change the styling associated with an element, such as the text color or underline. Here is an example that uses the style property:

<p><a href=".">Normal link</a></p>
<p><a href="." style="color: green">Green link</a></p>

A style attribute may contain one or more declarations, which are a property (such as color) followed by a colon and a value (such as green). When there is more than one declaration, they must be separated by semicolons, as in "color: red; border: none".

A lot of aspects of the document can be influenced by styling. For example, the display property controls whether an element is displayed as a block or an inline element.

This text is displayed <strong>inline</strong>,
<strong style="display: block">as a block</strong>, and
<strong style="display: none">not at all</strong>.

The block tag will end up on its own line since block elements are not displayed inline with the text around them. The last tag is not displayed at all—display: none prevents an element from showing up on the screen. This is a way to hide elements. It is often preferable to removing them from the document entirely because it makes it easy to reveal them again later.

JavaScript code can directly manipulate the style of an element through the element’s style property. This property holds an object that has properties for all possible style properties. The values of these properties are strings, which we can write to in order to change a particular aspect of the element’s style.

<p id="para" style="color: purple">
  Nice text

  let para = document.getElementById("para");
  console.log(; = "magenta";

Some style property names contain hyphens, such as font-family. Because such property names are awkward to work with in JavaScript (you’d have to say style["font-family"]), the property names in the style object for such properties have their hyphens removed and the letters after them capitalized (style.fontFamily).

Cascading styles

The styling system for HTML is called CSS, for Cascading Style Sheets. A style sheet is a set of rules for how to style elements in a document. It can be given inside a <style> tag.

  strong {
    font-style: italic;
    color: gray;
<p>Now <strong>strong text</strong> is italic and gray.</p>

The cascading in the name refers to the fact that multiple such rules are combined to produce the final style for an element. In the example, the default styling for <strong> tags, which gives them font-weight: bold, is overlaid by the rule in the <style> tag, which adds font-style and color.

When multiple rules define a value for the same property, the most recently read rule gets a higher precedence and wins. So if the rule in the <style> tag included font-weight: normal, contradicting the default font-weight rule, the text would be normal, not bold. Styles in a style attribute applied directly to the node have the highest precedence and always win.

It is possible to target things other than tag names in CSS rules. A rule for .abc applies to all elements with "abc" in their class attribute. A rule for #xyz applies to the element with an id attribute of "xyz" (which should be unique within the document).

.subtle {
  color: gray;
  font-size: 80%;
#header {
  background: blue;
  color: white;
/* p elements with id main and with classes a and b */
p#main.a.b {
  margin-bottom: 20px;

The precedence rule favoring the most recently defined rule applies only when the rules have the same specificity. A rule’s specificity is a measure of how precisely it describes matching elements, determined by the number and kind (tag, class, or ID) of element aspects it requires. For example, a rule that targets p.a is more specific than rules that target p or just .a and would thus take precedence over them.

The notation p > a {…} applies the given styles to all <a> tags that are direct children of <p> tags. Similarly, p a {…} applies to all <a> tags inside <p> tags, whether they are direct or indirect children.

Query selectors

We won’t be using style sheets all that much in this book. Understanding them is helpful when programming in the browser, but they are complicated enough to warrant a separate book.

The main reason I introduced selector syntax—the notation used in style sheets to determine which elements a set of styles apply to—is that we can use this same mini-language as an effective way to find DOM elements.

The querySelectorAll method, which is defined both on the document object and on element nodes, takes a selector string and returns a NodeList containing all the elements that it matches.

<p>And if you go chasing
  <span class="animal">rabbits</span></p>
<p>And you know you're going to fall</p>
<p>Tell 'em a <span class="character">hookah smoking
  <span class="animal">caterpillar</span></span></p>
<p>Has given you the call</p>

  function count(selector) {
    return document.querySelectorAll(selector).length;
  console.log(count("p"));           // All <p> elements
  // → 4
  console.log(count(".animal"));     // Class animal
  // → 2
  console.log(count("p .animal"));   // Animal inside of <p>
  // → 2
  console.log(count("p > .animal")); // Direct child of <p>
  // → 1

Unlike methods such as getElementsByTagName, the object returned by querySelectorAll is not live. It won’t change when you change the document. It is still not a real array, though, so you still need to call Array.from if you want to treat it like one.

The querySelector method (without the All part) works in a similar way. This one is useful if you want a specific, single element. It will return only the first matching element or null when no element matches.

Positioning and animating

The position style property influences layout in a powerful way. By default it has a value of static, meaning the element sits in its normal place in the document. When it is set to relative, the element still takes up space in the document, but now the top and left style properties can be used to move it relative to that normal place. When position is set to absolute, the element is removed from the normal document flow—that is, it no longer takes up space and may overlap with other elements. Also, its top and left properties can be used to absolutely position it relative to the top-left corner of the nearest enclosing element whose position property isn’t static, or relative to the document if no such enclosing element exists.

We can use this to create an animation. The following document displays a picture of a cat that moves around in an ellipse:

<p style="text-align: center">
  <img src="img/cat.png" style="position: relative">
  let cat = document.querySelector("img");
  let angle = Math.PI / 2;
  function animate(time, lastTime) {
    if (lastTime != null) {
      angle += (time - lastTime) * 0.001;
    } = (Math.sin(angle) * 20) + "px"; = (Math.cos(angle) * 200) + "px";
    requestAnimationFrame(newTime => animate(newTime, time));

Our picture is centered on the page and given a position of relative. We’ll repeatedly update that picture’s top and left styles to move it.

The script uses requestAnimationFrame to schedule the animate function to run whenever the browser is ready to repaint the screen. The animate function itself again calls requestAnimationFrame to schedule the next update. When the browser window (or tab) is active, this will cause updates to happen at a rate of about 60 per second, which tends to produce a good-looking animation.

If we just updated the DOM in a loop, the page would freeze, and nothing would show up on the screen. Browsers do not update their display while a JavaScript program is running, nor do they allow any interaction with the page. This is why we need requestAnimationFrame—it lets the browser know that we are done for now, and it can go ahead and do the things that browsers do, such as updating the screen and responding to user actions.

The animation function is passed the current time as an argument. To ensure that the motion of the cat per millisecond is stable, it bases the speed at which the angle changes on the difference between the current time and the last time the function ran. If it just moved the angle by a fixed amount per step, the motion would stutter if, for example, another heavy task running on the same computer were to prevent the function from running for a fraction of a second.

Moving in circles is done using the trigonometry functions Math.cos and Math.sin. For those who aren’t familiar with these, I’ll briefly introduce them since we will occasionally use them in this book.

Math.cos and Math.sin are useful for finding points that lie on a circle around point (0,0) with a radius of one. Both functions interpret their argument as the position on this circle, with zero denoting the point on the far right of the circle, going clockwise until 2π (about 6.28) has taken us around the whole circle. Math.cos tells you the x-coordinate of the point that corresponds to the given position, and Math.sin yields the y-coordinate. Positions (or angles) greater than 2π or less than 0 are valid—the rotation repeats so that a+2π refers to the same angle as a.

This unit for measuring angles is called radians—a full circle is 2π radians, similar to how it is 360 degrees when measuring in degrees. The constant π is available as Math.PI in JavaScript.

Using cosine and sine to compute coordinates

The cat animation code keeps a counter, angle, for the current angle of the animation and increments it every time the animate function is called. It can then use this angle to compute the current position of the image element. The top style is computed with Math.sin and multiplied by 20, which is the vertical radius of our ellipse. The left style is based on Math.cos and multiplied by 200 so that the ellipse is much wider than it is high.

Note that styles usually need units. In this case, we have to append "px" to the number to tell the browser that we are counting in pixels (as opposed to centimeters, “ems”, or other units). This is easy to forget. Using numbers without units will result in your style being ignored—unless the number is 0, which always means the same thing, regardless of its unit.


JavaScript programs may inspect and interfere with the document that the browser is displaying through a data structure called the DOM. This data structure represents the browser’s model of the document, and a JavaScript program can modify it to change the visible document.

The DOM is organized like a tree, in which elements are arranged hierarchically according to the structure of the document. The objects representing elements have properties such as parentNode and childNodes, which can be used to navigate through this tree.

The way a document is displayed can be influenced by styling, both by attaching styles to nodes directly and by defining rules that match certain nodes. There are many different style properties, such as color or display. JavaScript code can manipulate an element’s style directly through its style property.


Build a table

An HTML table is built with the following tag structure:


For each row, the <table> tag contains a <tr> tag. Inside of these <tr> tags, we can put cell elements: either heading cells (<th>) or regular cells (<td>).

Given a data set of mountains, an array of objects with name, height, and place properties, generate the DOM structure for a table that enumerates the objects. It should have one column per key and one row per object, plus a header row with <th> elements at the top, listing the column names.

Write this so that the columns are automatically derived from the objects, by taking the property names of the first object in the data.

Add the resulting table to the element with an id attribute of "mountains" so that it becomes visible in the document.

Once you have this working, right-align cells that contain number values by setting their style.textAlign property to "right".


<div id="mountains"></div>

  const MOUNTAINS = [
    {name: "Kilimanjaro", height: 5895, place: "Tanzania"},
    {name: "Everest", height: 8848, place: "Nepal"},
    {name: "Mount Fuji", height: 3776, place: "Japan"},
    {name: "Vaalserberg", height: 323, place: "Netherlands"},
    {name: "Denali", height: 6168, place: "United States"},
    {name: "Popocatepetl", height: 5465, place: "Mexico"},
    {name: "Mont Blanc", height: 4808, place: "Italy/France"}

  // Your code here

You can use document.createElement to create new element nodes, document.createTextNode to create text nodes, and the appendChild method to put nodes into other nodes.

You’ll want to loop over the key names once to fill in the top row and then again for each object in the array to construct the data rows. To get an array of key names from the first object, Object.keys will be useful.

To add the table to the correct parent node, you can use document.getElementById or document.querySelector to find the node with the proper id attribute.

Elements by tag name

The document.getElementsByTagName method returns all child elements with a given tag name. Implement your own version of this as a function that takes a node and a string (the tag name) as arguments and returns an array containing all descendant element nodes with the given tag name.

To find the tag name of an element, use its nodeName property. But note that this will return the tag name in all uppercase. Use the toLowerCase or toUpperCase string methods to compensate for this.

<h1>Heading with a <span>span</span> element.</h1>
<p>A paragraph with <span>one</span>, <span>two</span>

  function byTagName(node, tagName) {
    // Your code here.

  console.log(byTagName(document.body, "h1").length);
  // → 1
  console.log(byTagName(document.body, "span").length);
  // → 3
  let para = document.querySelector("p");
  console.log(byTagName(para, "span").length);
  // → 2

The solution is most easily expressed with a recursive function, similar to the talksAbout function defined earlier in this chapter.

You could call byTagname itself recursively, concatenating the resulting arrays to produce the output. Or you could create an inner function that calls itself recursively and that has access to an array binding defined in the outer function, to which it can add the matching elements it finds. Don’t forget to call the inner function once from the outer function to start the process.

The recursive function must check the node type. Here we are interested only in node type 1 (Node.ELEMENT_NODE). For such nodes, we must loop over their children and, for each child, see whether the child matches the query while also doing a recursive call on it to inspect its own children.

The cat’s hat

Extend the cat animation defined earlier so that both the cat and his hat (<img src="img/hat.png">) orbit at opposite sides of the ellipse.

Or make the hat circle around the cat. Or alter the animation in some other interesting way.

To make positioning multiple objects easier, it is probably a good idea to switch to absolute positioning. This means that top and left are counted relative to the top left of the document. To avoid using negative coordinates, which would cause the image to move outside of the visible page, you can add a fixed number of pixels to the position values.

<style>body { min-height: 200px }</style>
<img src="img/cat.png" id="cat" style="position: absolute">
<img src="img/hat.png" id="hat" style="position: absolute">

  let cat = document.querySelector("#cat");
  let hat = document.querySelector("#hat");

  let angle = 0;
  let lastTime = null;
  function animate(time) {
    if (lastTime != null) angle += (time - lastTime) * 0.001;
    lastTime = time; = (Math.sin(angle) * 40 + 40) + "px"; = (Math.cos(angle) * 200 + 230) + "px";

    // Your extensions here.


Math.cos and Math.sin measure angles in radians, where a full circle is 2π. For a given angle, you can get the opposite angle by adding half of this, which is Math.PI. This can be useful for putting the hat on the opposite side of the orbit.