Technical Drawing in Photonics Lesson 7 Primitive notions of geometric space. Projecting space onto a plane. Orthogonal projection. The double orthogonal projection system (descriptive geometry). Projection of a point. Projection of a line. Representation of a plane. TAMOP C-12/1/KONV project „Preparation of the concerned sectors for educational and R&D activities related to the Hungarian ELI project” Dr. Zsolt István Benkő

Technical drawing in Photonics Lesson 7 The primitive notions of geometric space are: points, lines, planes. All are undefined terms. The point is the most fundamental object: it has zero dimension (no size), it is represented by a dot and named by a capital letter. P point P

Technical drawing in Photonics Lesson 7 The line consists of infinitely many points. It is a one dimensional object. It has infinite length but has no width and height. It extends to infinitely far in two opposite directions. It is named usually by a small letter. A line segment may be named by the letters of the two endpoints. A B line e line segment AB e Every two distinct points denote a line and a line segment as well. If a third point fits to the line denoted by two points, then the points are called collinear.

Technical drawing in Photonics Lesson 7 A plane is an infinite set of points forming a connected flat surface. It extends infinitely far in all directions. It is a two dimensional object: it has infinite length and infinite width but no height. It is represented usually by a four sided figure and named by a capital letter. plane S A plane can be denoted by two intersecting lines or by a line and a non-fitting point or by three not collinear points.

Technical drawing in Photonics Lesson 7 Sometimes it is needed that a spatial object should be presented on paper or on a screen. In other words the three-dimensional space should be represented on a two-dimensional plane. This process is called projection. In a projection system projecting rays are used. (They are treated like idealized light rays.)

Technical drawing in Photonics Lesson 7 A projection ray originates from a point of the spatial object and where it hits the plane there would be the projection of the point. projection plane object point on object projected point on plane ray

Technical drawing in Photonics Lesson 7 There are many types of projection. The most common ones are the central projection and the parallel projection. In a central projection all the rays run into a single point called the center of projection. In a parallel projection all the rays are parallel. parallelcentral C

Technical drawing in Photonics Lesson 7 A special parallel projection is called the orthogonal projection. The rays are normal to the projection plane. In parallel projection if a spatial object is in a parallel position to the plane then the real measures can be obtained.

Technical drawing in Photonics Lesson 7 With a single projection plane only two dimensions could be recovered. If all spatial dimensions are needed (e.g. for production) then two projection planes are required. The most widely used is the double orthogonal projection system (other names: descriptive geometry; Monge geometry). It has two perpendicular projection planes. One is usually horizontal and the other is vertical. Their intersection is a line called the axis of projection or projection axis. A spatial point (P) is projected to the horizontal plane by a normal ray: it is the top-view or 1 st view (P’). The same point is projected to the vertical plane by a normal ray: it is the front-view or 2 nd view (P”).

Technical drawing in Photonics Lesson 7 The P, P’, P” points and a point of the axis denote a rectangle called the projection rectangle. (Two sides of it are the projecting rays.) The horizontal plane then rotated down around the axis into the vertical plane. The projection rectangle transforms into a line which is perpendicular to the axis. The top-view (1 st view) and the front-view (2 nd view) of the same spatial point is always connected by a line normal to the axis. Sometimes this line is called ”organizer” or ”organizer line”.

Technical drawing in Photonics Lesson 7 P P’ P” axis ray Basic arrangement

Technical drawing in Photonics Lesson 7 P’ P” axis Basic arrangement

Technical drawing in Photonics Lesson 7 P’ P” axis Basic arrangement

Technical drawing in Photonics Lesson 7 Basic arrangement

Technical drawing in Photonics Lesson 7 The projection planes are infinite. They divide the space into 4 quadrants. They are marked by roman numbers. The observer always views towards the axis. The closest quadrant to the observer is quadrant I. III III IV

Technical drawing in Photonics Lesson 7 P point is in quadrant I: P

Technical drawing in Photonics Lesson 7 Q point is in quadrant II: Q

Technical drawing in Photonics Lesson 7 R point is in quadrant III: R

Technical drawing in Photonics Lesson 7 S point is in quadrant IV: S

Technical drawing in Photonics Lesson 7 The points in the projection. P’ P” Q’ Q” R’ R” S’ S” x 1,2

Technical drawing in Photonics Lesson 7 Projecting a line e e’ e” x 1,2 e’ e”

Technical drawing in Photonics Lesson 7 Projecting a line

Technical drawing in Photonics Lesson 7 Trace on a plane x 1,2 f’ f” T1T1 T2T2

Technical drawing in Photonics Lesson 7 Special lines: rays e’ e” f’ f” 1 st ray2 nd ray x 1,2

Technical drawing in Photonics Lesson 7 Special lines: main lines e’ e” f’ f” 1 st main line2 nd main line x 1,2

Technical drawing in Photonics Lesson 7 Representation of a plane f’ f” g’ g” x 1,2

Technical drawing in Photonics Lesson 7 Representation of a plane t1t1 t2t2 x 1,2

Technical drawing in Photonics Lesson 7 References 1.Ocskó Gy., Seres F.: Gépipari szakrajz, Skandi-Wald Könyvkiadó, Budapest, Lőrincz P., Petrich G.: Ábrázoló geometria, Nemzeti Tankönyvkiadó Rt., Budapest, Pintér M.: AutoCAD tankönyv és példatár, ComputerBooks, Budapest, 2006