design | development Whether itʼs a 2D game involving sprites or a 3D game dealing with geometry, most of the action happens in 2D and 3D graphs respectively. Graphs in 2D games can be thought of or visualized as two-dimensional arrays. Just like this data structure, elementsʼ positions can be referenced to by their x- and y-coordinates. As for 3D games, graphs use the same concept. However, in this case, another axis is introduced: the z-axis, representing depth. And with graphs come coordinates. Coordinates are often used in conjunction with graphs to access locations, move an object, or get a direction, among other uses. In 2D graphs, coordinates have two key variables: one containing the x-position, with the other storing the y-position. Coordinates with three key variables contain the x- and y-position too, but they also use the z-position of the coordinate. These 3D coordinates are used with 3D graphs and are an extension of 2D ones, incorporating depth. Movement in video games makes direct use of coordinates. If in a platform game the player is moving forward 1 unit every frame, then the 1 unit is added to the playerʼs x-position in each frame. If the player is jumping and the change in vertical speed is 1 unit, 1 unit is added to the playerʼs y-position. What if the player is doing both simultaneously? 1 unit is added to both the x- and y-positions of the player. As for subtraction, it uses the same logic behind addition with coordinates. <7, 3> + <8, 4> = <15, 7> <7, 3> - <8, 4> = <-1, -1> Sometimes you may also need to multiply two or more coordinates. The same logic used in addition is used here too. The x- and y-positions of the initial coordinate are multiplied with the x- and y-positions of the second coordinate respectively. Division is precisely the same, but obviously coordinates are divided. The same concepts are used in diﬀerent arithmetic operations dealing with coordinates: the x-, y-, and z-components of coordinates canʼt be mixed - each coordinate should only be involved in operations with coordinates of the same type. <7, 3> × <8, 4> = <56, 12> <8, 4> ÷ <2, 2> = <4, 2> Mathematics and coordinates are considered the building blocks of video games. Without them, one canʼt manipulate objects, movement, or any other simple mechanisms. Having broad mathematical knowledge helps construct eﬃcient engines and understand better how games work. In the next article, weʼll take a look at how to calculate distance between two coordinates, and how to divide distance between two points proportionally. 14 EFG