**§ ****10 ****Helicoid**

[ Equation of a general helical surface ] The parametric equation of the known space curve is_{}

_{}

_{}The surface formed by the helical motion around the *z* -axis is called a general helical surface, and its equation is

Figure 7.21 |

_{}

where *h* is the pitch (or lead) .

[ Involute helical surface ] A surface formed by a known involute spiraling around the *z -axis on the **Oxy* plane is called an involute helical surface (Figure 7.21 ), and its equation is

_{}

Figure 7.22 |

where and are two parameters . " = constant" means different helixes on the surface, they have the same pitch *h* , but different helix angles; " = constant" means involutes at different positions ._{}_{}_{}_{}

[ Archimedes helical plane ] A helical plane formed by a helical motion of a straight line intersecting the *z* -axis and at an angle of intersection with the *Oxy* plane is called an Archimedes helical plane (Figure 7.22 ), and its equation is_{}_{}

_{}

Its intersection with the *Oxy* plane is an Archimedes spiral .

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