Isometric projection

Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and the angle between any two of them is 120 degrees. Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and the angle between any two of them is 120 degrees. The term 'isometric' comes from the Greek for 'equal measure', reflecting that the scale along each axis of the projection is the same (unlike some other forms of graphical projection). An isometric view of an object can be obtained by choosing the viewing direction such that the angles between the projections of the x, y, and z axes are all the same, or 120°. For example, with a cube, this is done by first looking straight towards one face. Next, the cube is rotated ±45° about the vertical axis, followed by a rotation of approximately 35.264° (precisely arcsin ​1⁄√3 or arctan ​1⁄√2, which is related to the Magic angle) about the horizontal axis. Note that with the cube (see image) the perimeter of the resulting 2D drawing is a perfect regular hexagon: all the black lines have equal length and all the cube's faces are the same area. Isometric graph paper can be placed under a normal piece of drawing paper to help achieve the effect without calculation. In a similar way, an isometric view can be obtained in a 3D scene. Starting with the camera aligned parallel to the floor and aligned to the coordinate axes, it is first rotated vertically (around the horizontal axis) by about 35.264° as above, then ±45° around the vertical axis. Another way isometric projection can be visualized is by considering a view within a cubical room starting in an upper corner and looking towards the opposite, lower corner. The x-axis extends diagonally down and right, the y-axis extends diagonally down and left, and the z-axis is straight up. Depth is also shown by height on the image. Lines drawn along the axes are at 120° to one another. The term 'isometric' is often mistakenly used to refer to axonometric projections, generally. There are, however, actually three types of axonometric projections: isometric, dimetric and trimetric. From the two angles needed for an isometric projection, the value of the second may seem counterintuitive and deserves some further explanation. Let’s first imagine a cube with sides of length 2, and its center positioned at the axis origin. We can calculate the length of the line from its center to the middle of any edge as √2 using Pythagoras' theorem . By rotating the cube by 45° on the x-axis, the point (1, 1, 1) will therefore become (1, 0, √2) as depicted in the diagram. The second rotation aims to bring the same point on the positive z-axis and so needs to perform a rotation of value equal to the arctangent of ​1⁄√2 which is approximately 35.264°. There are eight different orientations to obtain an isometric view, depending into which octant the viewer looks. The isometric transform from a point ax,y,z in 3D space to a point bx,y in 2D space looking into the first octant can be written mathematically with rotation matrices as: where α = arcsin(tan 30°) ≈ 35.264° and β = 45°. As explained above, this is a rotation around the vertical (here y) axis by β, followed by a rotation around the horizontal (here x) axis by α. This is then followed by an orthographic projection to the xy-plane: