Orthogonal is at a right angle.
Orthogonal is an adjective that is used to name something that is at an angle of 90º . This is a notion that, in the case of Euclidean spaces , is equivalent to the concept of perpendicularity .
We speak of orthogonal projection , on the other hand, to name the result of drawing all the perpendicular projecting lines on a certain plane. By performing this projection, a link is established between the points of the projecting component and the points of the projected element.
In addition to everything stated above, we can say that there are several different cases of orthogonal projection. Thus, however, among the most common and significant are the following two:
• The orthogonal projection of a segment.
• The orthogonal projection of a point.
The concept of orthogonal in Euclidean geometry
No less important is to highlight that, as a general rule, when talking about projection or orthogonal base it is done within the field of Euclidean geometry. This, also called parabolic or Euclidean, is that branch of study or discipline that is responsible for analyzing the properties of the geometric spaces where Euclid's axioms are fulfilled. That is, in three-dimensional space, on the real line or in the Euclidean plane.
The geometer and mathematician Euclid (325 – 265 BC) is the personality who shaped this discipline, which is based on various pillars such as the book he produced under the title “The Elements”. However, later figures such as Felix Klein made other contributions to it, through his “Erlangen Program”.
An orthogonal projection is obtained by drawing all the perpendicular projecting lines on a certain plane.
Example of such a projection
Suppose we want to make the orthogonal projection of a segment PR onto a line T. For this we will have to project the ends of PR through lines that are perpendicular to T , which will allow us to know the orthogonal projection of the segment on said line. The intersection between the projecting lines and T creates a new segment , which we could call MN . When the segment PR is parallel to the line T , the segment MN will be analogous to PR .
It can be said that the orthogonal layout is based on a right angle, developing in horizontal space and vertical space. This idea is not only applied in the field of geometry, but is also important in art . Artists must learn to work with orthogonality in an aesthetic sense so that the visual aspect of a painting is striking.
Orthogonal base vs. orthonormal base
It is common for confusion to occur between what is known as the orthogonal base and the orthonormal base. However, they are different and you have to know how:
• The first is a space as long as the vectors that form it have the particularity of being perpendicular two by two.
• The second, for its part, is the one that has a specific space whose base is orthogonal and its vectors also have the characteristic that they are unitary.
Circles can also be orthogonal when they intersect and, at a certain point, their respective tangents are perpendicular. With respect to a point of intersection, their radii will also be perpendicular.