CN102490522A

CN102490522A - Equal-arc ruler

Info

Publication number: CN102490522A
Application number: CN2011104443715A
Authority: CN
Inventors: 攸子铭
Original assignee: Individual
Current assignee: Individual
Priority date: 2011-12-27
Filing date: 2011-12-27
Publication date: 2012-06-13

Abstract

The invention discloses a tool for equally dividing a plane angle, and in particular relates to an equal-arc ruler. The ruler comprises a polar axis and an equal-arc line, wherein a polar coordinate equation of the equal-arc line is R=K/theta, R is radius, theta is radian, K is arc length, and the arc length K is a non-zero constant; the equal-arc line is an equal-arc line with theta of 0 to 2pi; and the arc length K is the non-zero constant, the radius R and the radian theta is an inverse proportional function according to the polar coordinate equation R=K/theta, the equal division of an angle is converted into the amplification of the radius R by the corresponding multiple, and only the radius is needed to be amplified by the corresponding multiple and a point on the equal-arc line is selected and connected with a pole when one angle is equally divided, so that the equally-divided angle can be obtained. The equal-arc ruler is convenient to use and easy to operate and learn, the plane angle can be easily and arbitrarily equally divided by means of compasses and the equal-arc ruler, and the problem that the accuracy of equally dividing the angle is reduced due to complex repeated conversion and a large number of equal fractions is solved.

Description

Deng the arc chi

Technical field

The present invention relates to a kind of construction with ruler and compasses instrument, particularly a kind of branch instruments such as plane angle of five equilibrium plane angle arbitrarily.

Background technology

In construction with ruler and compasses that any five equilibrium of line segment is fairly simple; Realize easily, be insoluble problem in the construction with ruler and compasses and plane angle is carried out any five equilibrium always, once was called as one of " big how much difficult problems in ancient times three " like the trisection of an angle; Though people have also found some to utilize the method for ruler sub-multiple angle at present; But the common complicated steps of these methods is implemented cumbersomely, and precision is not high.

Like Chinese patent number be: the utility model patent of CN87208343U discloses a kind of plane angle halver.This plane halver is for a circle or the same trapezoidal formation plane geometric shape of combining of hemicycle at least; Circle or at least its diameter of hemicycle overlap with a trapezoidal base (short base); Circle or at least its semicircle girth of hemicycle with another trapezoidal base (promptly long base) appearance etc.; Engraving the evenly divided line of identical five equilibrium respectively on the semi arch with on the long base,, divide quarter thin more to realize the mutual conversion of circular arc and line segment; Sub-multiple angle is accurate more, circle or at least its radius of hemicycle can less than, be equal to or greater than trapezoidal height.

Utilize one of above-mentioned plane angle halver five equilibrium less than 180 ° plane angle operation as follows: the summit with this angle is the center of circle, is that radius is drawn arc with the radius of halver, obtains the pairing circular arc in this angle (promptly being central angle at this moment); With compasses or divider on the semi arch of halver, measure this section arc length; On long base, intercept corresponding length along path through corresponding line segment, this line segment of five equilibrium obtains Along ent; Find corresponding Along ent on the circular arc of measuring on the halver conversely along corresponding line segment; This section circular arc that has been five equilibrium with compasses or divider five equilibrium on the pairing circular arc in angle, links up the five equilibrium of promptly having realized the angle with Along ent and summit.

The disclosed plane angle halver of above-mentioned Chinese patent; Its principle is based on the plane geometry theorem: the central angle and the corresponding with it contained radian number or the number of degrees of circular arc equate; At first convert the process of a plane angle of five equilibrium to five equilibrium corresponding line segment; Behind the branches such as line segment, the given plane angles of branch such as another mistake correspondence.Though it has realized the five equilibrium of plane angle, above-mentioned plane angle halver wants angle step of five equilibrium more; When sub-multiple angle, must angle be converted into arc length earlier, find the line segment of corresponding arc length again; Another mistake just can be found out the plane angle of five equilibrium at last, troublesome poeration to finding corresponding arc length behind the five equilibrium line segment; In the routine drawing, usually expend a lot of times.And its can only five equilibrium smaller or equal to 180 ° angle, as want five equilibrium greater than 180 ° plane angle, must be earlier this angle being halved obtains two equal plane angles less than 180 °; As above each angle of five equilibrium then, but the five equilibrium of realizing like this is the original twice that requires, and obtain the angle that needs; Also will be again with the angle addition that waits behind the branch, its complicated steps, the sub-multiple angle that obtains through multiple conversions; Its precision no doubt can be influenced, causes the sub-multiple angle angle to occur than large deviation.

Summary of the invention

It is a kind of simple to operate that the present invention provides, but the branch instruments such as plane angle of High Accuracy and Divide plane angle.

Branch instruments such as plane angle of the present invention are a kind of arc chi that waits, and it comprises pole axis and wait camber line that the polar equation that waits camber line is R=K/ θ; Wherein R is a radius, and θ is a radian, and K is an arc length; Arc length K is the constant of non-zero, and the is said camber line of Denging is the camber line that waits of 0-2 π for radian θ.

The arc chi that waits of the present invention; θ such as R=K/ such as its camber line such as grade mainly is a relation of using for reference angle and radius in the polar coordinates; Be (a is a constant) that spiral of Archimedes R=a θ accomplishes; In spiral of Archimedes, radius and angle are directly proportional and concern that utilizing spiral of Archimedes to pass through its radius of five equilibrium can the five equilibrium plane angle.The present invention is with reference to the equation R=a θ of spiral of Archimedes; Set up polar equation R=K/ θ (K is the constant of a non-zero); R θ is the computing formula of arc length in polar coordinate system; Because R θ=K, thus polar equation R=K/ θ is an arc length be a constant, radius and angle inversely proportional etc. camber line.

The arc chi that waits of the present invention comprises pole axis, and according to the camber line that waits of polar equation R=K/ θ foundation, wherein R is a radius; θ is a radian, and K is an arc length, and arc length K is the constant of a non-zero; Because arc length K is the constant of a non-zero; According to polar equation R=K/ θ, radius R and radian θ are the inverse proportion function, the five equilibrium at angle are converted into the corresponding multiple amplification of radius R.When angle of five equilibrium, earlier pole axis, angular alignment limit are aimed in a limit at angle; The another side at angle is given a bit with waiting camber line mutually, utilizes compasses will wait intersection point and the distance R between the limit on the camber line on pole axis, to amplify the multiple (if with branch n equal portions such as angles, just R being enlarged n doubly) of wanting five equilibrium; Utilizing compasses is that radius is drawn circular arc with the line segment after amplifying; Camber line such as give, connect intersection point and limit after intersecting then, promptly obtain the angle of five equilibrium.

Utilize the inverse proportion function five equilibrium at angle to be converted into the multiple amplification of radius R; Only need when wanting angle of five equilibrium the point that the amplification that radius is done corresponding multiple is chosen etc. on the camber line then is connected the angle that can obtain five equilibrium with limit; Easy to use; Easy simple to operate just can be easily with any five equilibrium of plane angle by compasses and chi such as arc such as grade.Avoid because of the repeatedly conversion of complicacy and isodisperse more for a long time, the problem that the precision of five equilibrium angle reduces.

Description of drawings

Fig. 1 is a sketch map of the present invention.

Fig. 2 is a kind of use sketch map of invention.

Fig. 3 is a kind of embodiment of the present invention.

Fig. 4 is another kind of embodiment of the present invention.

Fig. 5 is the another kind of sketch map that uses of the present invention.

The specific embodiment

Below in conjunction with accompanying drawing, the present invention is further described.

In polar coordinate system, it is exactly spiral of Archimedes that angle and radius the most directly concern, its equation is R=a θ (a is a constant); In spiral of Archimedes, radius R and angle θ relation in direct ratio, utilizing spiral of Archimedes to pass through its radius R of five equilibrium can five equilibrium plane angle θ; Radius is divided into little line segment, is that radius is drawn circular arc and spiral of Archimedes intersects with the line segment behind the five equilibrium, intersection point is connected with limit makes sub-multiple angle; But its limitation is arranged by the spiral of Archimedes sub-multiple angle; Under a fairly large number of situation of five equilibrium, radius is divided into very little line segment, is that radius is drawn circular arc and spiral of Archimedes intersects with it; Precision is difficult to guarantee, causes the sub-multiple angle precise decreasing that obtains.

The present invention is with reference to the equation R=a θ of spiral of Archimedes; Set up polar equation R=K/ θ; K is the constant of a non-zero; R θ is the computing formula of arc length just in polar coordinate system, in this equation because R θ=K, so polar equation R=K/ θ is an arc length be a constant, radius and angle inversely proportional etc. camber line.

Like Fig. 1, the arc chi that waits of the present invention, it comprises pole axis and waits camber line; Polar equation Deng camber line is R=K/ θ, and wherein R is a radius, and θ is the radian at angle; K is an arc length, and arc length K is the constant of a non-zero, and the is said camber line of Denging is the camber line that waits of 0-2 π for radian θ.Because arc length K is the constant of a non-zero, according to polar equation R=K/ θ, radius R and radian θ are the inverse proportion function, the five equilibrium at angle are converted into the amplification of the corresponding multiple of radius R.Like figure, wait the arc chi can be the elongated rectangular shape shape, wait the zone between camber line and the pole axis to empty, pole location can punch or use transparent material to make, and has been convenient to observe the angle of treating five equilibrium, the convenient drawing.

Like Fig. 2, sketch map when having provided angle θ of arc chi 3 five equilibriums such as a kind of utilization is aimed at pole axis with the limit of angle θ earlier; The angular alignment limit, the another side at angle is given an A mutually with waiting camber line, utilizes compasses to wait and on pole axis, amplifies 3 times (as if with branch n equal portions such as angles apart from r between intersection point A and the limit O on the camber line; Just R is enlarged n doubly), utilizing compasses is that radius is drawn arc with the line segment 3r after amplifying, and waits camber line to give a B with friendship; Connect intersection points B and limit O then, promptly obtain angle θ/3 of five equilibrium.R=K/ θ can obtain Rsin θ=K, and (sin θ/θ), θ is tending towards 0 o'clock sin θ=θ, so Rsin θ=K; Be that θ leveled off to 0 o'clock, wait camber line R=K/ θ to be gradually to the straight line of Rsin θ=K, thus wait the prolongation that camber line can be unlimited, even littler its precision of angle of five equilibrium can not reduce yet.

The arc chi that waits of the present invention; Utilize the inverse proportion function five equilibrium at angle to be converted into the multiple amplification of radius R; Only need when wanting angle of five equilibrium the point that the amplification that radius is done corresponding multiple is chosen etc. on the camber line then is connected the angle that can obtain five equilibrium with limit; Easy to use, easy simple to operate just can be easily with any five equilibrium of plane angle by compasses and chi such as arc such as grade.And, realize when isodisperse is big, only needing the five equilibrium of plane angle radius is amplified corresponding multiple by the inverse proportion function, avoid causing the precise decreasing of five equilibrium angle because of five equilibrium repeatedly.

Like Fig. 3; Fig. 4 has provided two kinds of different chis such as arc such as grade; Because the relation of purposes and ruler length; It chooses radian respectively is camber line such as between π/18-pi/2 (angle is 10 °-90 °) and the π/18-π (angle is 10 °-180 °) a section, and as the measurement category that waits the arc chi, the scope at the angle that it can five equilibrium is π/18-pi/2 (angle is 10 °-90 °) and π/18-π (angle is 10 °-180 °).

The above waits the arc chi, is convenient the use, according to the standard angle mark angle index is arranged waiting on the camber line, the angle index that is marked, and it is the center of circle with the limit, is that the zero degree limit marks with the pole axis.And, more convenient in order the arc chi such as to use to draw in construction with ruler and compasses, is waiting the tip edge labeled standards length scale of arc chi, it also can work as the ruler use simultaneously.

The above waits the another kind of usage of arc chi is the picture regular polygon; Like Fig. 5 is arc chis such as more convenient use; A triangular marker is arranged on the pole axis that waits the arc chi, and limit is to wait the radius length of camber line R=K/ θ when θ=2 π (angle is 360 °), the i.e. position of R=K/2 π to the distance expression of first graduation mark; As the distance of a unit, on pole axis, begin to mark scale simultaneously from limit.Because the radius when R=K/2 π of unit representes θ=360 °, when enlarging R with integral multiple, θ dwindles with integral multiple simultaneously.So, be the drawn circular arc of radius with n the scale value that marks, just in time be the corresponding circular arc in limit of positive n limit shape.

With reference to figure 5 is the just heptagonal sketch map of picture; At first the distance with 7 scales of limit to the of waiting the arc chi is that radius is drawn circle; To wait arc chi limit to aim at the center of circle then, rotation waits the arc chi, the corresponding circular arc in each limit of positive heptagon on the intercepting circumference; Connect each circular arc cut-point at last, promptly obtain positive heptagon.

Claims

1. one kind is waited the arc chi, it is characterized in that, comprises pole axis and waits camber line, and the polar equation that waits camber line is R=K/ θ, and wherein R is a radius, and θ is a radian, and K is an arc length, and arc length K is the constant of non-zero, and is said, and to wait camber line be the camber line that waits of 0-2 π for radian θ.

2. chi such as arc such as grade according to claim 1 is characterized in that, the said camber line that waits is the arbitrary section camber line such as grade of 0-2 π for radian θ.

3. the arc chi that waits according to claim 1 and 2 is characterized in that, to wait the camber line subscript to be marked with the limit be the center of circle, be the angle index on zero degree limit with the pole axis.

4. the arc chi that waits according to claim 1 and 2 is characterized in that, on pole axis, begins to mark the scale that equates at interval from limit, and the distance between adjacent scale is K/2 π.