Helix belongs to space curve, which has many forms such as cylindrical helix, conical helix, etc.stayArchitectureCylindrical helix is the most commonly used in mechanical engineering[1-2]。
There are all kinds of poeticcurveHelix is one of the special types.The term "spiral" comes from Greek, which originally means“Coiling”Or "winding".For example, a plane spiral is a curve formed by winding outward one by one from a fixed point[3]。
More than 2000 years ago, Archimedes, an ancient Greek mathematician, studied helices.Famous mathematicianDescartes In 1638, the logarithmic spiral was first described, and the analytic formula of the spiral was listed.What's more interesting is that the Swiss mathematician Yagu Bernoulli asked someone to carve a snail shaped log spiral on his tombstone before he died, and humorously wrote the epitaph of "I will revive after changing in the original way"[3]。
spider webIt is a widely distributed and impressive spiral structure in nature.The structure of the spider web fully demonstrates what a wonderful life a spider is with a wonderful spiral concept[3]。
The leaves of plantain are also spirally arranged, with the included angles of 137 degrees, 30 degrees and 38 degrees.This arrangement of leaf order can make the leaves get the maximum lighting and good ventilation.In fact, the arrangement of plant leaves on the stem is generally spiral.In addition, the arrangement of sunflower seeds on the plate is also spiral[3]。
Human hair grows obliquely from the scalp hair follicles. It forms a vortex along a certain direction, which is called hair rotation, and there are right and left rotations.In fact, hair twirls are hair twirls that grow on the body surface and can make hair grow in a certain direction.In wild mammals, cyclotron has the function of protecting itself and adapting to the environment.It can make the rain flow in a certain direction, just like putting on a coir raincoat;They are closely arranged to avoid the bites of harmful insects;In addition, it has a good thermal insulation effect.Although these functions of human hair have degenerated to a negligible extent, their forms have remained[3]。
Helix generated by special movement
1. An ant crawls outward along the radius in the center of a uniformly rotating record at a constant speed, and as a result, the ant itself draws a spiral line[3]。
2. Bats fly down from high places, following the path of space spiral -- conical spiral[3]。
3. Chasing fleeing enemy ships or seizing smuggled ships on the sea, sometimes we have to follow the spiral path[3]。
4. Some orbits of stars are also spirals.National Astronomical Observatory of JapanNaoto Nakai, who has observed and studied the gas density in the middle of the Milky Way for three years, believes that the Milky Way is in a spiral shape, that is, stars expand outward in a spiral shape with the center of a circle[3]。
classification
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Helix, or spiral.It can be divided into 2D spiral and 3D spiral by dimension[3]。
Where a and b are real numbers.Changing parameter a is equivalent to rotating the spiral, while parameter b controls the distance between two adjacent curves[3]。
The plane Cartesian coordinate equation of Archimedes spiral is[3]：
Transformation from Polar Coordinate System to Cartesian Coordinate System[3]：
2. Fermat spiral[3]
Fermat spiral is a kind of equiangular spiral, and the expression is:
Fermat spiral
3. Equiangular spiral[3]
Equiangular spiral refers to the form:
Spiral of[3]。
Equiangular spiral
4. Hyperbolic spiral[3]
Polar coordinate equation r θ=c, where c is a constant.In rectangular coordinate system,x=c*cosθ\θ，y=c*sinθ\θ。Therefore, the visible asymptote is: y=c[3]。
Hyperbolic spiral
5. Intracircular spiral[3]
A moving small circle is inscribed in a fixed large circle. During the rolling process of the small circle, the track formed by the previous fixed point is called the spiral inside the circle.This point will have different trajectories with the different ratio of the radius of the two circles[3]。
In particular, when the radius of the small circle is equal to half of the large circle, the locus of each point of the small circle is a diameter of the large circle;When the radius of the small circle is equal to one fourth of the large circle, the track formed is a star line (see Figure 3)[3]。
When the radius ratio of big circle to small circle is changed, the trajectory formed is shown in Figure 1 below[3]：
Figure 1 and Figure 2
Figure 3 and Figure 4
6. Interlocking spiral[3]
Interlocking spiral, also called Lituus spiral in English (as shown in Figure 5 below), is derived from the parameterization and interaction of three conic curves in high school mathematics[3]。
The parameterization of three conic curves in high school mathematics is all spirals in the form of r ˆ 2 * θ=k.Interlocking spiral r ˆ 2 θ=k meets: there is a horizontal asymptote y=0[3]。
Figure V
Note: in the formula, r is the radius, θ is the included angle, and k is the slope[3]。
7. Corona spiral[3]
KeNu spiral is a graph drawn by functions in the field of mathematics[3]。
3D Helix
Cylindrical helix[3]
The front projection of circular helix is a sine curve, and the horizontal projection is a circle.According to the rotation direction of the moving point, the helix can be divided into left helix and right helix[3]。
Cylindrical helix
When a moving point moves along the straight generatrix of the cylindrical surfaceUniform linear motionWhen the generatrix rotates uniformly around the axis of the cylindrical surface at the same time, the track of the moving point is a cylindrical helix.As shown in Figure 6[1][4-7]。
Figure 6
Among them, the linear distance that the moving point moves along the axis when rotating for one cycle is called lead (or pitch)[1][4-7]。
Helix can be right-handed or left-handed[1][6-7]：
1. When the axis is vertical, the visible part of the helix rises from left to right, which is called the right helix[1][6-7]；
2. Left rotation When the axis is vertical, the visible part of the helix rises from right to left, it is called the left helix[1][6-7]。
Therefore, the three elements to determine the cylindrical helix are: cylindrical surface diameter, lead, and direction of rotation[1]。
Projection practice
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As shown in Figure 7, assume that the cylinder axis is perpendicular to the H plane, the diameter and lead of the cylinder surface are known, and the spiral is right-handed.So the drawing process of the spiral is[1][7]：
1. Make two side projections of the cylindrical surface, and divide the lead reflected by the circle in the horizontal projection and the front projection into the same equal parts;Then, number the equal points on the circumference according to the right rotation rule, and number the equal points on the lead line from bottom to top[1][7]；
2. Make horizontal lines and vertical lines through the equal points with the same number of front projection and horizontal projection respectively, and they intersect each other to get a series of intersection points 1 ', 2', 3 ',..., 8'. Connect these points smoothly in turn, and distinguish the visibility to get the front projection of the spiral line. The horizontal projection coincides with the accumulated projection (circumference) of the cylinder surface[1][7]。
