Lesson 2
Surface Sketching:
Sphere, Cylinder and
Quadrics
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Surface Sketching: Spheres, Cylinders, and Quadric Surfaces, Slides of Calculus for Engineers
A lesson on surface sketching, focusing on spheres, cylinders, and quadric surfaces. It includes illustrative problems and examples to help students understand how to sketch these surfaces in a three-dimensional system. The lesson covers finding the center and radius of a sphere, determining the equation of a sphere, and describing surfaces based on their equations. Additionally, it introduces different types of quadric surfaces and provides activities for practice. This material is suitable for students learning multivariable calculus and analytic geometry, offering a practical approach to visualizing and representing three-dimensional shapes. The document aims to enhance students' ability to analyze and sketch various surfaces, which is crucial for advanced mathematical studies.
Typology: Slides
2024/2025
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Lesson 2
Surface Sketching:
Sphere, Cylinder and
Quadrics
OBJECTIVE:
- At the end of the lesson, the student must be able to:
- Sketch other surfaces like sphere, cylinders and
quadrics in three dimensional system.
Illustrative
Problem
Illustrative
Problem
Examples:
- Find the equation of a sphere of radius 3 centered at (2, 3, 9). Given : Radius ( r ) = 3 ; C(2, 3, 9) Solution : (x – x 0 ) 2 + (y – y 0 ) 2 + (z – z 0 ) 2 = r (x – 2) 2 + (y – 3) 2 + (z – 9) 2 = 3
- Describe the surface whose equation is given. x 2 + y 2 + z 2 + 4x + 6y + 10z – 26 = 0 Solution : x 2 + y 2 + z 2 + 4x + 6y + 10z – 26 = 0 (x 2 + 4x)+ (y 2 + 6y)+( z 2 +10z) = 26 (x 2 + 4x + 4)+ (y 2 + 6y + 9)+( z 2 +10z + 25) = 26 + 4 + 9 + 25 (x + 2) 2 + (y + 3) 2 + ( z + 5) 2 = 64 sphere with center C (-2, -3, -5) and r = 8
SURFACES : Cylinder
Illustrative Problem
Illustrative Problem
PARABOLIC CYLINDER
Illustrative
Problem
PARABOLIC CYLINDER
Illustrative
Problem
- z 2 = y – 1
PARABOLIC CYLINDER
SURFACES : Quadric
Common Types of Quadric Surfaces
1. Ellipsoid
2. Elliptic Paraboloid
3. Elliptic Cone
4. Hyperboloid of One Sheet
5. Hyperboloid of Two Sheets
6. Hyperbolic Paraboloid
QUADRIC SURFACE
nd
Type of Quadric Surface
rd