Cone Cutting Activity.ni.sir.ian.pls.dont
- 1. Cone Cutting Activity Christian S. Abasta, LPT Special Science Teacher 1
- 2. Double Right Circular Cone • Suppose a fixed vertical line ℓ is intersected by a line 𝓂 at a fixed point 𝒫 and at an angle 𝛼. If the line 𝓂 rotates around ℓ in such a way that the angle 𝛼 remains constant, then the surface generated is called the Double Right Circular Cone.
- 4. Parts of a Double Right Circular Cone 1. Axis • the fixed line ℓ 2. Vertex • fixed point 𝒫 3. Generator or Generatrix • the line 𝓂 that rotates about the vertex 𝒫 4. Vertex Angle • fixed angle 𝛼 • the acute angle between the generator 𝓂 and the axis ℓ
- 5. Parts of a Double Right Circular Cone 5. Base • Circle • The axis ℓ is perpendicular to and passes through the center of the base. 6. Directrix • the circumference (or perimeter) of the base 7. Nappe • lateral surface of a cone • Upper Nappe (above the vertex) • Lower Nappe (below the vertex)
- 6. Conic Section •a section or curve obtained by intersecting a right circular cone by a plane. •The plane is usually called the cutting plane.
- 7. Important angles: 𝜶 and 𝜷 •Note that 𝜶 is the acute angle between the generator 𝓂 and the axis ℓ •Let 𝜷 be an acute angle made by the cutting plane with the axis ℓ of the cone.
- 8. Circle •𝜷 = 𝟗𝟎° •The cutting plane is perpendicular to the axis of the cone
- 9. Ellipse •𝜶 < 𝜷 •The cutting plane intersects only one nappe to form a bounded curve
- 10. Parabola •𝜶 = 𝜷 •The cutting plane intersects only one nappe to form an unbounded curve. •The cutting plane is parallel to the generator 𝓂 or to an element of the cone. •Each line composing the surface of the cone is called an element.
- 11. Hyperbola Case 1: 𝜶 > 𝜷 The cutting plane intersects the axis ℓ to form two unbounded curves. Case 2: 𝜷 does not exist. The cutting plane is parallel to the axis ℓ to form two unbounded curves. • The cutting plane intersects both nappes to form two unbounded curves.
- 12. Breakdown of Points Task Point Circle 20 Ellipse 20 Parabola 20 Hyperbola 20 Overall Presentation 20 Total 100