Lecture 7: The Metric problems

The Main Menu اPrevious اPrevious Next The metric problems 1- Introduction 2- The first problem 3- The second problem 4- The third problem a- Rotation b- Affinity 6- Examples

Metric problems deal with :, the angles between two straight lines or two planes True lengths, true shapes, perpendicularity, the angles between two straight lines or two planes Or a straight line and a plane and the rotation of Planes. least The right angle is projected into a right angle iff at least one of its legs is parallel to the plane of projection. The Main Menu اPrevious اPrevious Next A B C AiAi BiBi CiCi 1- Introduction Theorem (1)

The Main Menu اPrevious اPrevious Next x 12 C2C2 B1B1 C1C1 A2A2 B2B2 A1A1. A2A2 B2B2 C2C2 A1A1 B1B1 C1C1. The horizontal projection of the angle ABC is right angle The vertical projection of the angle ABC is right angle T.L AB//  1 AB//  2

The Main Menu اPrevious اPrevious Next A3A3 B3B3 C3C3 x 12 A2A2 B2B2 C2C2 A1A1 B1B1 C1C1 x 13. The side projection of the angle ABC is right angle T.L

Example (1) : Given the side AB of a square ABCD and the horizontal Projection of a straight line m on which the side BC lies Represent this square by its two projections The Main Menu اPrevious اPrevious Next x 12 A2A2 B2B2 A1A1 B1B1 m1m1

The Main Menu اPrevious اPrevious Next x 12 A2A2 x 13 B2B2 A1A1 B1B1 A3A3 B3B3 T.L. K1K1 m1m1 K3K3. K2K2 [K] // T.L //. [C] C1C1 C2C2 D1D1 D3D3 // m3m3

The normal n through a given point M to a given plane. h f n M A straight line n is perpendicular to a plane if it is perpendicular to two intersecting straight lines h and f lying in the plane. h is taken a horizontal straight line and f is taken a frontal straight line in the given plane. We use THEOREM (2) to represent the normal n. The Main Menu اPrevious اPrevious Next To construct a straight line through a given point and perpendicular to a given plane. 2- The first problem Theorem (2)

n 1 passes thr, M 1 and is normal to h ρ. i) the plane is given by its traces. The Main Menu اPrevious اPrevious Next n 2 passes thr, M 2 and is normal to v ρ. x 12 M2M2 M1M1 n2n2 n1n1

ii) the plane is given by two intersecting str. Lines a and b. We use a horizontal str. Line h and frontal str. Line f in the plane. The Main Menu اPrevious اPrevious Next x 12 a2a2 b2b2 a1a1 b1b1 M2M2 M1M1 h2h2 1\1\ 1 2 2\2\ h1h1 f1f1 f2f2 34 3\3\ 4\4\ n2n2.. n1n1.

To construct a plane ( normal plane) through a given point and perpendicular to a given straight line. i) The normal plane is determined by two straight lines h and f. The Main Menu اPrevious اPrevious Next h2h2 f1f1 x 12 m2m2 m1m1 M2M2 M1M1 h1h1. f2f2. 3- The second problem

The Main Menu اPrevious اPrevious Next ii) The normal plane is determined by its traces. x 12 m2m2 m1m1 M2M2 M1M1 h2h2. h1h1 v1v1 v = v 2..

i) The rotation of a plane about its horizontal trace till it coincides with the Horizontal plane Π 1. The Main Menu اPrevious اPrevious Next x 12 M2M2 M1M1. // zMzM [M] * * (M) 4- The third problem ( The rotation)

Is one to one correspondence between points or straight lines. It is defined by an axis o called the axis of affinity and a direction d called the direction of affinity and two corresponding points M and M \. M d A If a point A is given, to find A \. Q join QM \ and draw a segment parallel to d from A cutting QM \ in the point A \. Join AM Find Q on o The Main Menu اPrevious اPrevious Next o M\M\ A\A\ 4- The third problem ( Affinity)

The Main Menu اPrevious اPrevious Next x 12 M2M2 M1M1. // zMzM [M] * * (M) A2A2 A1A1 (A)

ii) The rotation of a plane about its vertical trace till it coincides with the Vertical plane Π 2. The Main Menu اPrevious اPrevious Next x 12 M2M2 M1M1. // yMyM [M] * * (M) A2A2 A1A1 (A)

x 12 M2M2 M1M1. // zMzM [M] * * (M) A1A1 (A) v) The rotation of a plane about a horizontal straight line h till it coincides with the horizontal plane Π passing through the horizontal straight line h. The Main Menu اPrevious اPrevious Next {.

The Main Menu اPrevious اPrevious Next x 12 M2M2 M1M1. // yMyM [M] * * (M) A2A2 (A) vi) The rotation of a plane about a frontal straight line f till it coincides with the frontal plane passing through the frontal straight line h.. {

Given two straight lines a and b intersecting in a point M. Find the angle < (a, b) and represent its bisector. (M) The Main Menu اPrevious اPrevious Next x 12 a2a2 b2b2 M2M2 M1M1 b1b1 a1a1 h2h2 h1h1 { { A2A2 A1A1 B2B2 B1B1 (b) (a) (R) R1R1 R2R2

Represent a square ABCD lying in a given plane. If the vertical projections of A and C are given. Hence find a point E such that: AE = BE = CE= DE = 6 cms. The Main Menu اPrevious اPrevious Next x 12 C2C2 A2A2 + +

C2C2 A2A2 C1C1 A1A1 (A) (C). (M) M1M1 M2M2 D1D1 B1B1 B2B2 6 cm MA E. The Main Menu اPrevious اPrevious Next // E1E1 K2K2 K1K1 The true length of MK to get the direction of true length of n. [K] * E2E2 [E] * D2D2 (D) (B) n1n1 n2n2 // + +

Given two planes by its traces find the dihedral angle between the two planes and. M n n\n\ The Main Menu اPrevious اPrevious From M: where x 12 M2M2 M1M1 n2n2 n1n1 n\2n\2 n\1n\1 h2h2 h1h1 { {. (M) (n \ ) (n)