◇ 매개변수표현

1. 삼각함수 : x = Cos[q], y = Sin[q]

ParametricPlot[{Cos[q], Sin[q]}, {q, 0, 2Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_3.gif]

2. 원 (Circle) : x = a Cos[q], y = a Sin[q]

a = 2 ; ParametricPlot[{a Cos[q], a Sin[q]}, {q, 0, 2Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_6.gif]

3. 타원 : x = a Cos[q], y = b Sin[q]

a = 2 ; b = 1 ; ParametricPlot[{a Cos[q], b Sin[q]}, {q, 0, 2Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_9.gif]

4. 쌍곡선 : x = a Sec[q], y = b Tan[q]

a = 2 ; b = 1 ; ParametricPlot[{a Sec[q], b Tan[q]}, {q, 0, 2Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_12.gif]

5. 포물선 : x = 2 a t, y = a t^2

a = 2 ; ParametricPlot[{2 a t, a t^2}, {t, -2, 2}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_15.gif]

6. 파선 (Cycloid) : x = a (t - Sin[t]), y = a (1 - Cos[t])

a = 2 ; ParametricPlot[{a (t - Sin[t]), a (1 - Cos[t])}, {t, -2Pi, 2 Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_18.gif]

7. 바퀴안파선 (HypoCycloid) : x = (a - b) Cos[t] + b Cos[(a - b)/a t], y = (a - b) Sin[t] - b Sin[(a - b)/a t]

a = 2 ; b = 1 ; ParametricPlot[{(a - b) Cos[t] + b Cos[(a - b)/a t], (a - b) Sin[t] - b Sin[(a - b)/a t]}, {t, -2Pi, 2 Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_21.gif]

8. 바퀴위파선 (EnCycloid) : x = (a + b) Cos[t] - b Cos[(a + b)/b t], y = (a + b) Sin[t] - b Sin[(a + b)/b t]

a = 2 ; b = 1 ; ParametricPlot[{(a + b) Cos[t] - b Cos[(a + b)/b t], (a + b) Sin[t] - b Sin[(a + b)/b t]}, {t, 0, 2Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_24.gif]

9. 성망형 (Astroid) : x = Cos[t]^3, y = Sin[t]^3

ParametricPlot[{Cos[t]^3, Sin[t]^3}, {t, 0, 2Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_27.gif]

10. 심장형 (Cardioid) : x = 2b Cos[t] - b Cos[2t], y = 2b Sin[t] - b Sin[2t]

b = 1 ; ParametricPlot[{2b Cos[t] - b Cos[2t], 2b Sin[t] - b Sin[2t]}, {t, 0, 2Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_30.gif]

11. 데카리스엽선 (Descartes Lenf) : t≠ -1, x = 3t/(1 + t^3), y = 3t^2/(1 + t^3)

ParametricPlot[{3t/(1 + t^3), 3t^2/(1 + t^3)}, {t, -10, 10}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_33.gif]

12. 아르키메데스의나선 (Spiral) : x = q Cos[q], y = q Sin[q]

ParametricPlot[{q Cos[q], q Sin[q]}, {q, 0, 10Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_36.gif]

13. 연주형 (Comliscate) : x = ±Cos[q] Cos[q], y = ±Cos[2q] Sin[q]

ParametricPlot[{Cos[q] Cos[q], Cos[2q] Sin[q]}, {q, -1 Pi, 1Pi}, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_39.gif]

◇ 공간도형

1. 구 : x = a Sin[u] Sin[v], y = a Sin[u] Cos[v], z = a Cos[u]

a = 2 ; ParametricPlot3D[{a Sin[u] Sin[v], a Sin[u] Cos[v], a Cos[u]}, {u, 0, 2Pi}, {v, 0, 2 Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_42.gif]

2. 나팔곡면 (Pseudo Sphere) : x = a Sin[u] Cos[v], y = a Sin[u] Sin[v], z = a (Cos[u] + Log[Tan[u/2]])

a = 2 ; ParametricPlot3D[{a Sin[u] Cos[v], a Sin[u] Sin[v], a (Cos[u] + Log[Tan[u/2]])}, {u, 0.2Pi, Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_45.gif]

3. 상나선 (Circular Helix) : x = a Cos[t], y = a Sin[t], z = b t (a>0, b≠0)

a = 2 ; b = .1 ; ParametricPlot3D[{a Cos[t], y = a Sin[t], z = b t}, {t, 0, 3Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_48.gif]

4. 나선면 (Helicoid) : x = v Cos[u], y = v Sin[u], z = b u

b = 2 ; ParametricPlot3D[{v Cos[u], v Sin[u], b u}, {u, 0, 2Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_51.gif]

5. 현수면 (Catenoid) : x = a Cosh[v] Cos[u], y = a Cosh[v], z = a v

a = 2 ; ParametricPlot3D[{a Cosh[v] Cos[u], a Cosh[v], a v}, {u, 0, 2 Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_54.gif]

6. 윤활면 (Torus) : x = (a + b Cos[u]) Cos[v], y = (a + b Cos[u]) Sin[v], z = b Sin[u] (a>b)

a = 2 ; b = 1 ; ParametricPlot3D[{(a + b Cos[u]) Cos[v], (a + b Cos[u]) Sin[v], b Sin[u] }, {u, 0, 2 Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_57.gif]

7. 원주면 (Circulorcylinder) : x = a Sin[u], y = a Cos[u], z = v

a = 2 ; ParametricPlot3D[{a Sin[u], a Cos[u], v }, {u, 0, 2 Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_60.gif]

8. 타원면 (Eilisoid) : x = a Cosh[u] Cos[v], y = a Cosh[u] Sin[v], z = b Sinh[u]

a = 2 ; b = 1 ; ParametricPlot3D[{a Cosh[u] Cos[v], a Cosh[u] Sin[v], b Sinh[u] }, {u, 0, 2 Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_63.gif]

9. 일엽쌍곡선 (One Sheet Hyperboloid) : x = a Cosh[u] Cos[v], y = b Cosh[u] Sin[v], z = c Sinh[u]

a = 3 ; b = 2 ; c = 1 ; ParametricPlot3D[{a Cosh[u] Cos[v], b Cosh[u] Sin[v], c Sinh[u] }, {u, 0, 2Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_66.gif]

10. 이엽쌍곡선 (Two Sheet Hyperboloid) : x = a Sinh[u] Sin[v], y = b Sinh[u] Sin[v], z = c Cosh[u]

a = 3 ; b = 2 ; c = 1 ; ParametricPlot3D[{a Sinh[u] Sin[v], b Sinh[u] Sin[v], c Cosh[u] }, {u, 0, 2Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_69.gif]

11. 타원포물면 (Elliptic Paraboloid) : x = a u Cos[v], y = b u Sin[v], z = u^2

a = 3 ; b = 2 ; ParametricPlot3D[{a u Cos[v], b u Sin[v], u^2 }, {u, 0, 2Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_72.gif]

12. 쌍곡포물면 (Hyperboloid Paraboloid) : x = a u Sec[v], y = b u Tan[v], z = u^2

a = 3 ; b = 2 ; ParametricPlot3D[{a u Sec[v], b u Tan[v], u^2 }, {u, 0, 2Pi}, {v, 0, 2Pi}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_75.gif]

13. 뫼비우스띠 (Moobius Band) : x = (a - v Sin[u/2]) Cos[u], y = (a - v Sin[u/2]) Sin[u], z = v Cos[u/2]

a = 3 ; ParametricPlot3D[{(a - v Sin[u/2]) Cos[u], (a - v Sin[u/2]) Sin[u], v Cos[u/2] }, {u, -Pi, Pi}, {v, 0, 1}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_78.gif]

14. 원숭이안장곡면 (Monkey Saddle) : x = u, y = v, z = u^3 - 3u v^2

ParametricPlot3D[{u, v, u^3 - 3u v^2}, {u, -1, 1}, {v, -1, 1}, Boxed→False, AspectRatio→Automatic] ;

[Graphics:HTMLFiles/index_81.gif]