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Cartesian Coordinates in Space


If any figure has three dimensions than it has three co-ordinates i.e. X, Y and Z axis. Cartesian coordinates in space is denoted by Point ‘P’ which contains three Real Numbers that indicate the positions of the perpendicular projections from the point to three Perpendicular Lines which are called axis. If the Cartesian coordinate contains three points x, y, z then the point ‘x’ is belongs to X axis, point ‘y’ belongs to Y axis and point ‘z’ belongs to Z axis. We can easily write that Cartesian coordinate P = (x, y, z). If we have two, three dimension co-ordinates and we want to calculate the distance between them then we use pythagoras theorem to calculate distance between them. Let’s assume we have two Cartesian coordinates (x1, y1, z1) and (x2, y2, z3) then the distance ‘d’ between two co-ordinates is
=√[ (x2 - x1)2 + (y2 – y1) + (z2 – z1)2]
Now to understand we will take an example of two Cartesian points and find distance between them. We have two Cartesian point A (3, 4, 8) and B (2, 3, 6) then according to formula distance
D = √[(2- 3)2 + (3 – 4 )2 + (6 – 8)2],
After putting the values in the formula we get
D = √[(-1)2 + (-1)2 + (-2)2],
D = √(6) = 2.44.
So the distance between two co-ordinates ‘A’ and ‘B’ is 2.44. If three dimension Cartesian co-ordinates ‘P’ contains three points x = 0, y = 0 and z = 0 then it means that ‘P’ is a point which is situated at the origin O (0, 0, 0) or point ‘P’ and Origin ‘O’ are same and the distance between any point to ‘P’ is same as origin to that point.

Orthogonal Projections

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Orthogonal projections are defined as the way of representing a three dimensional object into two dimensional representations. It represents parallel form in the projection where all projection lines are obtained as orthogonal and that is to the projection plane. It can be further classified or divided in different types of orthographic projections. First is termed as multilevel orthogonal projection and second is axonometric projection.

Orthographic projection is also used in lenses and such types of lenses are known as telecentric lenses. Multiview type of orthographic or orthogonal projections are defined as projections in which six object pictures are formed and these pictures are produced with projection planes which are parallel to coordinate axis of the objects. Two schemes are used for positioning the views that are first angle projection and third angle projection. These projections can be easily understood by the help of machine drawing. In mathematical terms when any angle is projected two notions are considered that are defined as notion of perpendicularity as well as the notion of distance.

When a angle is projected say 'θ' from any Point to a line then it forms a perpendicular over the line at which the angle is projected or we can also say that it forms a perpendicular by which angle is projected. These perpendiculars possess some distance which can be determined.

Suppose there are two lines AB and AC having a common point A and forming an angle θ between them. Let CD is the perpendicular drawn on AB from point C, this perpendicular shows or represents the projection of line AC on line AB and the distance AD that is the distance of point A from the perpendicular drawn is represented by the cosine function as distance AB is equals to AC Cos θ and AB and AC vectors are represented in modulus form.