How to Rotate a Triangle on a Grid


Two-dimensional rotations come up often in digital image production and game programming. As objects move around in a virtual world they can both translate, and rotate. In two dimensions, a rotation is defined by an axis, the point about which the rotation takes place, and the angle of rotation. To rotate a triangle, you rotate the three vertices that define the triangle.

Triangle in the Grid

  • Points on a two-dimensional grid are written (x, y). A triangle has three such points, one defining each vertex. For instance, we could label these p1 = (x1, y1), p2 = (x2, y2) and p3 = (x3, y3).

Rotations around the Origin

  • If you rotate a point p by an angle θ, you get a new point p’ = (x’, y’) where x’ = x∙cos(θ) - y∙sin(θ) and y’ = y∙cos(θ) + x∙sin(θ). To rotate a triangle, you apply this process to p1, p2 and p3.

Rotations around an Arbitrary Axis

  • To rotate around an arbitrary axis, you translate the axis to the origin, taking the point p with it, perform the rotation as above, then undo the translation. Let P = (X, Y) be the axis. Translate both the axis and the point to the origin by subtracting P. This gives a new axis, P = P - P = (0, 0), and a new point p = (x, y) = p-P = (x - X, y - Y). Perform the rotation as usual on p. Finally, undo the initial translation by adding P to get x’ = X + (x - X)∙cos(θ) - (y - Y)∙sin(θ) and y’ = Y + (y - Y)∙cos(θ) + (x - X)∙sin(θ). Repeat this process for each point.


  • Let a triangle be defined by the points p1 = (1,1), p2 = (2,3) and p3 = (4,2). Rotate the triangle around the point P = (3,-1) by 23 degrees. Using the formula, p1 - P = (-2,2) and p1’_x = 3 + (-2)∙cos(23) - 2∙sin(23) = 0.3775, and p1’_y = -1 + 2∙cos(23) + (-2)∙sin(23) = 0.0595. Transforming the other points in a similar way, we have the new points: p1’ = (0.3775, 0.0595), p2’ = (0.5166, 2.2913) and p3’ = (2.7483, 2.1522).

Related Searches


  • Photo Credit Dario Lo Presti/iStock/Getty Images
Promoted By Zergnet


You May Also Like

Related Searches

Check It Out

Can You Take Advantage Of Student Loan Forgiveness?

Is DIY in your DNA? Become part of our maker community.
Submit Your Work!