Lecture 8: The circle & the sphere

The Sphere 1- The orthogonal projection of the sphere 2- Representation of the sphere in Monge’s projection 3- Examples

The Sphere The sphere is projected as a circle with radius r = true length To draw a circle you must Know 1- The center. 2- The radius. s r

The Sphere Basics 1- العمودى على المماس من نقطة التماس يمر بالمركز (محل هندسى للمركز) s s A B m

2- العمودى على الوتر من المنتصف يمر بالمركز (محل هندسى للمركز) s A B N

n 1 passes thr, M 1 and is normal to h ρ. Draw a line  a plane. n 2 passes thr, M 2 and is normal to v ρ. x 12 M2M2 M1M1 n2n2 n1n1

Draw a plane   M, and  line m. x 12 m2m2 m1m1 M2M2 M1M1 h2h2. h1h1 v1v1 v = v 2..

Example (6): A sphere passing through A (-4, 4.5, 1.5), B(0,7.5, 6.5) and its center  m:{(3,5,3), (4,5,3)}, Construct Monge’s projection of the sphere s A B K m, L:S 1- نوجد K منتصف AB 2- نرسم مستوى  يحتوى K و  AB 3-   m = S 4- r= AS =BS

x 12 0 z A1A1 S2S2 A2A2 B2B2 K2K2 m 2 L:S 2 Example (6): A sphere passing through A (-4, 4.5, 1.5), B(0,7.5, 6.5) and its center  m:{(3,5,3), (4,5,3)}, Construct Monge’s projection of the sphere B1B1 m 1 L:S 1 K1K1 vv hh S1S1 B2B2 S2S2  y BS T.L v 

Example (7): Let m:{A(-1,1,2), B(-1,4.5,5.5)} be a line and let  (2,1,1) be a plane and  be a sphere touching , its center S  m, its radius = 2.5 cms. Construct Monge’s projection of the sphere 1- نرسم مستوى  // المستوى  ويبعد عنه   m = S s   m, L:S 2.5  3- نرسم الكرة بمعلومية المركز و نصف القطر

Example (7): Let m:{A(-1,1,2), B(-1,4.5,5.5)} be a line and let  (2,1,1) be a plane and  be a sphere touching , its center S  m, its radius = 2.5 cms. Construct Monge’s projection of the sphere x 12 z A1A1 A2A2 B2B2 L:S 2 B1B1 L:S 1 vv hh 2.5 x 13 33 m 2 m 1 v  hh 33 B3B3 A3A3 S3S3 S1S1 S2S2

The circle 1- The orthogonal projection of the circle 2- How to draw an ellipse 3- Representation of a circle in Monge’s projection 4- Examples

The orthogonal projection of the circle K   onto any proj. plane  i is an ellipse With major axis parallel to ρ ∩  i with length equal to the diameter of the circle. 1- The orthogonal projection of the circle

Given the major axis and a point M  the ellipse? (find the length b of the semi minor axis ). 1- نرسم عمودا على المحور الاكبر من منتصفه. 2- نركز فى النقطة المعلومهM وبفتحة تساوى نصف طول المحو ر الاكبر نرسم قوسا يقطع اتجاه المحور الاصغر فى نقطة K 3-نصل MK فيقطع المحور الاكبر فى N فيكون MN هو نصف المحور الاصغر. AA\A\. M. N b K. 2- How to draw an ellipse M

α Π 1. We must know: 1- The plane of the circle. 2- The center of the circle. 3- The radius of the circle x 12 A2A2 B2B2 C2C2 D2D2 D1D1 B 1 = A 1 C1C1 // s2s2. r c 1 d 1 = true length = 2r. A 2 B 2 = 2r. The major axis A 2 B 2 //

α Π 2. x 12 A1A1 B1B1 C1C1 D1D1 s1s1. D2D2 B 2 = A 2 C2C2 // r C 2 D 2 = true length = 2r. The major axis A 1 B 1 // A 1 B 1 = 2r.

x 12 0 z y s1s1 A1A1 B1B1 // r s2s2 A2A2 B2B2 r C2C2 D2D2 C1C1 D1D1

x 12 0 z y s1s1 A1A1 B1B1 // 3 s2s2 A2A2 B2B2 3 C2C2 D2D2 C1C1 D1D1 Example 3: Represent a circle lying in a plane ρ( -7, 7, 5.5). Its center (1, 3.5, ?) and its radius has a length 3 cms.