Intersection Points Between a Line and a Circle

Most of the time people want to know if a circle is colliding with a line or not (you can do that by finding the shortest distance and check if it's equal or less to the width of the radius). But what if you need to know exactly where the line is intersecting the circle? First we need to take into account the equation of the line and the circle: y = m x + b (x - Cx)2 + (y - Cy)2 = r2 But before replacing the Y in the circle equation we're going to expand it, using this formula: (a - b)2  =  a2 - 2ab + b2 So: (x - Cx)2 + (y - Cy)2 = r2  <=>  x2 - 2xCx...

Read more...

Polygon Collision Detection

Detecting collisions for regular polygons is relatively simple. A square will always have 4 sides, a pentagon will always have an internal angle of 108º and so on. Their shapes and properties are always the same, no matter their size or position. But what if we're dealing with irregular polygons? They can assume an infinite number of shapes, sides and angles. They're impossible to predict! The first thing we need to learn is how we tell if a point is inside a polygon or not. There are several techniques, but I'm going to use the Jordan Curve Theorem. This theorem basically says that a closed...

Read more...

Finding a Point in the Line at a Specific Distance

Imagine that you have a line, a point (p1) belonging to that line and you need to find a 2nd point (p2), also belonging to the line, that's at a specific distance (d) from the 1st point. We could try to find the circle that has p1 as the center and d as the radius and check where the circle intersects with the line. But I want to try a different method that uses the line's slope and trigonometry. First we have to take into account the equation that define the slope (m) of a line: m =  y2 - y1          x2 - x1 If we take a closer look, and...

Read more...

Shortest Distance Between a Point and a Line

Knowing if 2 objects are touching each other is only a part of collision detection. There are occasions when knowing how far each object is from each other is just as important. Finding the shortest distance between 2 points is done with the Distance Formula. And the same goes for finding the distance between a point and a circle (you just need to check the distance between the point and the center and subtract the radius). But the distance between a point and a line (or between other polygons defined with straight lines) is found through a different method. We can draw an infinite number...

Read more...

Simple Line Collision Detection

It might not be obvious at first, but Line Intersections are actually very handy. For example, you can use them to define complex polygon shapes (octagons, stars, etc) and from there calculate the collision between them! But in order to determine where the lines collide with each other, we first need the equation of a straight line: y = m x + b m =  y1 - y2           x1 - x2 So to determine any given point of a line we need to know its slope (m) and its y-intercept (b). Then all we need to do is give a X coordinate to obtain the corresponding...

Read more...

Simple Circle Collision Detection

Circle are pretty popular. From basketballs and billiard balls to bullseye targets and frisbees, there are plenty of games where circles play an important role. But unless we're dealing with soap bubbles, we need a way to tell the computer when the circles are colliding. If we take the Equation of the Circle (with "r" being the radius and "C" the center of the circle) we can easily determine if a point (P) is inside the circle: ( Px - Cx )2 + ( Py - Cy )2 <= r2 But if we want to detect the collision between 2 circles, it's more intuitive to calculate the distance between the centers...

Read more...

Simple Rectangle Collision Detection

We use collision detection all the time. Grabbing a cup of coffee, scratching our nose, sitting on a chair... even when walking our body detects the presence of other objects and surfaces outside of its own. For computers it's a bit different. They can't detect anything unless we tell them how. (Unrotated) rectangle collision is one of the simplest ones. If any point (P) isn't between the coordinates of both axis of the rectangle (R), then it's not colliding: if ( Px >= R1x  AND  Px <= R2x  AND      Py >= R1y  AND  Py <= R2y){  ...

Read more...

Finding the length of a line (Distance Formula)

I've once heard a story on how Pythagoras came across his Pythagorean Theorem. One day he was walking on a tiled floor, much like the image above. As he was looking at the pattern it formed, he noticed something amazing. The sum of the squares of the cathetus was equal to the square of the hypotenuse! Or in mathematical terms:  a2 + b2 = c2 But how is this tied to finding the length of a line? This formula applies to all right-angled triangles, the same type of triangles drawn by the Cartesian coordinate system. So if we have the length of the X and Y we calculate the length of the...

Read more...

Finding the X and Y projections of a line

The traditional screen is like a sheet of paper. You can draw things in perspective and give the illusion of depth, but ultimately everything is represented in a 2D plane. If you tried to move your head around to see behind an object on screen, the image would remain the same. The computer uses Cartesian coordinates to describe the positions of the objects along an X and Y axis, like in this example (the direction of the Y-axis is reversed though). But what if we have a line of a specific length and at a certain angle? How do we know the X and Y coordinates of both ends of the line? That's...

Read more...

Converting Radians into Degrees and vice-versa

We use angles all the time. How high the sun is in the sky, how much a wheel turns, how much our arm rotates or how much we turn our head... If it wasn't for that measurement of the circular motion, we wouldn't be able to understand how the stars move or predict their positions during the course of the year (imagine how many people who would get their horoscopes wrong). We're more used to measure angles in degrees. Pretty much everyone knows if you spin around yourself you make a 360º turn to return to the original position. Or if you stretch out your arms perpendicularly they make a 90º...

Read more...