Elliptical system derivative parameter fast calculation disc and operation method thereof

Abstract

The invention discloses an elliptic system derivative parameter fast calculation disc which comprises a center ring, and the center ring is sequentially connected with a middle ring and a side ring through a mortise and tenon structure. A pointer is fixed to the center of the center ring through a fixing shaft. Scales are arranged on the central ring, the middle ring and the side rings; the pointer is in a rectangular shape with an opening on one side, and the opening end of the pointer clamps the circle center of the center ring through a fixing shaft. The method overcomes the defects that manual calculation is large in labor amount and large in repeated labor amount, simplifies deep theoretical tedious calculation and numerous and jumbled data searching into simple operation of one-to-one-pair, has the advantages of wonderful conception, scientific design, simple structure, simple and convenient operation, easy popularization, high applicability, low cost, convenience in carrying and the like, and is suitable for popularization and application. Good popularization and application prospects are realized.

Description

Technical Field

[0001] This invention relates to the field of elliptical system derived shape calculation, specifically to a quick abacus for elliptical system derived shape parameters and its calculation method. Background Technology

[0002] Elliptical derivative shapes refer to a series of planar figures or solid geometric shapes formed by extending the characteristics of an ellipse or the equation of an ellipse, including circles, sectors, ellipses, spheres, spherical wedges, ellipsoids, ellipsoidal wedges, tori, torus wedges, etc.

[0003] Elliptical systems, as a commonly used type of geometric shape, have wide applications in daily life and in fields such as astronomy, engineering, optics, aerospace, geography, and machining, and their parameters are frequently calculated. However, because the calculation of their parameters often involves advanced functions such as differentials and integrals, the calculation process is extremely complex and tedious, causing great inconvenience or difficulties for the general public and professionals (especially frontline workers).

[0004] The calculation of parameters for existing elliptic derivative shapes (including circles, sectors, ellipses, spheres, spherical wedges, ellipsoids, ellipsoidal wedges, tori, torus wedges, etc.) is mainly done manually or by computer. It often involves advanced functions such as differentials and integrals, and the calculation process is extremely complex and tedious. Computer calculation also requires computer equipment, mathematical software, and corresponding environmental conditions, which brings great inconvenience or difficulties to users.

[0005] Patent document CN119673453A discloses an intelligent nursing safety management skin management calculation board, which includes a multi-layer concentric circle chart module, a calculation model and intelligent analysis algorithm module, a touch operation interface, a data recording and export module, and an implementation module. This calculation board uses a multi-layer concentric circle chart module to display skin stages, clinical manifestations, nursing measures, and prevention strategies through different colors and layers, making it convenient for nursing staff to intuitively obtain information. The system integrates a skin health score calculation model, a treatment plan recommendation algorithm, a nursing effect prediction model, and a risk warning algorithm. Based on multiple skin health indicators, it accurately assesses the patient's skin condition and updates nursing suggestions in real time.

[0006] Patent application CN13253795A discloses a formula fertilization quick calculation tool and its application method as a soil fertilization measurement and formulation technology. It includes three parts: a pointer, a compound inlaid combination turntable, and a scale. The compound inlaid combination turntable has a disc in the middle, and a fixed shaft is installed at the center of the disc. The pointer is installed on the fixed shaft. The two sides of the compound inlaid combination turntable are engraved with scales.

[0007] Therefore, providing a fast abacus for elliptical system-derived shape parameters and its calculation method that focuses on practicality, innovative carriers, avoids difficulties and simplifies complexity is a problem worthy of research. Summary of the Invention

[0008] To address the shortcomings of the existing technology and facilitate the work of users, this invention provides a quick abacus for ellipsoidal shape parameters and its calculation method, based on mathematical formulas, quick calculation principles, and algorithm theory, and tailored to the needs of daily life and related work.

[0009] The objective of this invention is achieved as follows:

[0010] An elliptical-derived shape parameter abacus includes a central ring, which is connected to a middle ring and a side ring in sequence by a mortise and tenon structure; a pointer is fixed to the center of the central ring by a fixed shaft.

[0011] The central ring, middle ring, and side ring are provided with scales.

[0012] The pointer is rectangular with one side open, and one end of the opening is clamped to the center of the central ring through a fixed shaft.

[0013] The central ring is a multi-ring structure with detachable mortise and tenon joints.

[0014] Parameter calculations for circles and spheres: The calculation methods for the circumference and area of ​​a circle, and the surface area and volume of a sphere, are directly based on the provided information, including the following steps (taking the circumference of a circle as an example):

[0015] (1) Locate the r-ring and ⊙L-ring on surface A of the elliptical system derived shape parameter quick calculation disk and align their starting points (red line positions);

[0016] (2) Find the scale point A corresponding to the radius value of the circle in the r ring; when it is not an integer, estimate its position using interpolation method;

[0017] (3) Find the scale point B on the ⊙L ring that corresponds to point A on the r ring. The value of this scale point is the circumference of the circle, and the unit is the same as the radius unit. When it is between two scale points, use the interpolation method to estimate its value.

[0018] The calculation of the area of ​​a circle, the surface area of ​​a sphere, and the volume of a sphere can be performed using the methods described above.

[0019] The steps for calculating the circumference of an ellipse are as follows:

[0020] (1) According to the aforementioned method for calculating the circumference of a circle, find point B on the circle with radius b as the minor semi-axis of the ellipse on the ⊙L ring, i.e. r = b;

[0021] (2) The starting point of the loop is aligned with point B of the ⊙L loop;

[0022] (3) In Find the scale point C corresponding to the difference between the major semi-axis a and the minor semi-axis b of the ellipse on the ring, i.e., ab.

[0023] (4) Find on the ⊙L ring Point D corresponds to point C on the ring; the scale value here represents the circumference of the ellipse.

[0024] The calculation of the area of ​​an ellipse and the surface area of ​​a torus requires two steps: equivalent parameter calculation and verification (taking the surface area calculation of a torus as an example).

[0025] (1) In ① Locate point E on the ring, corresponding to the inner radius R of the ring; ② Locate point F, which corresponds to the outer radius r of the torus. ① Ring and ② The meaning and function of rings are exactly the same and they can be interchanged;

[0026] (2) ② The starting point of the ring is aligned ① Point E of the ring, at this time ② Point F of the ring is at ① The scale value of the corresponding point G on the ring is the equivalent parameter d of the toroidal surface R*r;

[0027] (3) Find the position H of the d value in the r ring, and the scale value of the corresponding position I on the ⊙S ring is the area value of the ellipse.

[0028] The calculation method for the area of ​​an ellipse is the same as that for the surface area of ​​a torus, except that the inner radius R and outer radius r of the torus are replaced with the major semi-axis a and minor semi-axis b of the ellipse. Simply replace the ring with a ⊙S ring.

[0029] The calculation of torus volume is similar to that of torus surface area, and the calculation steps are as follows:

[0030] (1) First, find the toroidal surface area calculation method described above. ① Point G on the ring, corresponding to the equivalent parameter (d) of the toroidal surface R*r;

[0031] (2) Then ② The starting point of the ring is aligned ① Point G of the ring;

[0032] (3) In ② Find point J on the ring corresponding to the outer radius r. At this time... ① The scale value of point K on the ring, which corresponds to point J, is the equivalent parameter f of R*r2;

[0033] (4) Locate point L on the r-ring corresponding to the value of f. The scale mark at point M on the ring, corresponding to point M, represents the volume of the ring surface.

[0034] The calculation methods for ellipsoidal volume and torus volume are the same; only the R*r*r of the torus needs to be replaced with the a*b*c of the ellipsoid. Simply replace the ring with a ●V ring.

[0035] The method for calculating the surface area of ​​an ellipsoid is as follows:

[0036] (1) First, find the ellipsoidal volume using the aforementioned method. ① Point K on the ring, corresponding to the equivalent parameter f of the ellipsoid a*b*c;

[0037] (2) Locate points N and P on the r-ring corresponding to the f value and 2 / 3 of the f value, and align point N with point K. At this point... ① The scale value at point Q on the ring, corresponding to point P, is (abc). 2 / 3 The equivalent parameter g;

[0038] (3) Find the position R corresponding to the g value on the r ring. At this time, the scale of the position S on the ●S ring corresponding to the R point is the surface area value of the ellipsoid.

[0039] Parameter calculations for sector-shaped, spherical, ellipsoidal, and toroidal wedges:

[0040] The calculations of sector arc length, sector area, spherical wedge sphere area, spherical wedge volume, ellipsoidal wedge sphere area, ellipsoidal wedge volume, torus wedge arc area, and torus wedge volume are based on the circumference of a circle, the surface area of ​​a sphere, the surface area of ​​an ellipsoid, and the surface area of ​​a torus, respectively. Only in the equivalent parameter step is it necessary to utilize... ③ Further adjustments to the equivalent parameters of the ring are needed. Specifically, taking the calculation of the sector area as an example, the steps are as follows:

[0041] (1) First in ③ Locate the corresponding central angle mark T on the ring; ③ Unlike other rings, the ring starts at 360° and the degrees are arranged in descending counterclockwise order;

[0042] (2) Align the starting point of ring r with point T, and... ③ Mark the starting point of the ring on the r-ring as point U, and find the position at 1 / 2U and mark it as point V;

[0043] (3) In ① Locate point W on the sector radius scale on the ring and align point W with the scale. ③ The starting point of the loop, at which point V is located ① The scale value at point X on the ring is the equivalent radius h of the sector;

[0044] (4) Find the corresponding position Y of h value on the r ring. At this time, the scale value of the corresponding position of Y on the ⊙S ring is the area value of the sector.

[0045] The equivalent parameter adjustment methods for the remaining seven shape parameters are the same and can be followed accordingly. It is particularly important to note that: the equivalent parameter adjustment coefficient for the sector arc length (the ratio of arc UV / arc OU on the r-ring, see the figure above) is 1; the equivalent parameter adjustment coefficients for the sector area, spherical wedge sphere area, and toroidal wedge arc area are 1 / 2; the equivalent parameter adjustment coefficients for the spherical wedge volume, ellipsoidal wedge volume, and toroidal wedge volume are 1 / 3; and the equivalent parameter adjustment coefficient for the ellipsoidal wedge sphere area is 2 / 3.

[0046] Positive and beneficial effects: This invention overcomes the shortcomings of manual calculation, which involves a large amount of labor and repetitive work. It simplifies the complex calculations and data searches of profound theories into a simple "one-to-one" operation. It has the advantages of ingenious conception, scientific design, simple structure, easy operation, easy promotion, strong applicability, low cost and portability, and has good prospects for promotion and application. Attached Figure Description

[0047] Figure 1 Schematic diagram of circles and sectors;

[0048] Figure 2 . is a schematic diagram of an ellipse;

[0049] Figure 3 A schematic diagram of a sphere and a spherical wedge;

[0050] Figure 4 Schematic diagram of the ellipsoid (left) and ellipsoidal wedge (right);

[0051] Figure 5 Schematic diagram of the torus (left) and torus wedge (right);

[0052] Figure 6 Schematic diagram of the scaled toroidal rings of .r, L, and S;

[0053] Figure 7 Schematic diagram of the abacus structure with elliptical system derived shape parameters;

[0054] Figure 8 A diagram illustrating the steps for calculating the circumference of a circle;

[0055] Figure 9 A schematic diagram illustrating the steps for calculating the circumference of an ellipse;

[0056] Figure 10 A schematic diagram illustrating the steps for calculating the surface area of ​​a torus;

[0057] Figure 11 Schematic diagram of the steps for calculating the volume of a torus;

[0058] Figure 12 Schematic diagram of the steps for calculating the surface area of ​​an ellipsoid Figure 1 ;

[0059] Figure 13 Schematic diagram of the steps for calculating the surface area of ​​an ellipsoid Figure 2 ;

[0060] Figure 14 Schematic diagram of the steps for calculating the area of ​​a sector Figure 1 ;

[0061] Figure 15 Schematic diagram of the steps for calculating the area of ​​a sector Figure 2 . Detailed Implementation

[0062] The invention will be further explained below with reference to specific implementation examples:

[0063] Elliptical system derived shapes include circles, sectors, ellipses, spheres, spherical wedges, ellipsoids, ellipsoidal wedges, tori, and toroidal wedges. The following definitions apply to elliptical system derived shapes:

[0064] I. Definition of Terminology:

[0065] (a) Circles and sectors

[0066] A circle is the set of points in a plane that are equidistant from a fixed point (the center) by a fixed distance (radius). The portion bounded by two radii and an arc is called a sector, and the angle between the two radii is called the central angle (θ). The standard equation of a circle is:

[0067] x 2 +y 2 =r 2

[0068] Circles and fan shapes, such as Figure 1 As shown:

[0069] (II) Ellipse

[0070] like Figure 2 As shown in the diagram, an ellipse is the locus of a moving point on a plane whose sum of distances to two fixed points (foci) is a constant (greater than the focal length). Its mathematical expression and standard equation are as follows:

[0071] |PF1|+|PF2|=2a(2a>|F1F2|)

[0072]

[0073] (III) Sphere and Spherical Wedge

[0074] A sphere is a geometric solid formed by rotating a semicircle around its diameter. The center of the semicircle is the center of the sphere, and the radius of the semicircle is the radius of the sphere. The area enclosed by two circular planes passing through the same diameter of the sphere and the spherical surface between them is called a wedge, and the angle r between the two hemispheres is called the wedge angle. A schematic diagram of a sphere and a wedge is shown below. Figure 3 as follows:

[0075] The standard equation of a sphere is:

[0076] r 2 =x 2 +y 2 +z 2

[0077] (iv) Ellipsoid and Ellipsoid Wedge

[0078] In a spatial rectangular coordinate system, by the standard equations The geometric solid defined by (a>0, b>0, c>0) is called an ellipsoid (see diagram below). When a = b = c, it is a sphere. When a = b ≠ c, or a ≠ b = c, or a = c ≠ b, it is an ellipsoid of revolution. When a ≠ b ≠ c, it is a triaxial ellipsoid. The geometric solid formed by the rotation of two half-planes passing through the same semi-axis (or diameter) and the ellipsoid (or sphere) they enclose is called an ellipsoidal wedge, and the angle between the two half-planes is called the ellipsoidal wedge angle. (ellipsoid) Figure 4-1 ) and ellipsoidal wedge ( Figure 4-2 ) Schematic diagram as follows Figure 4 As shown below:

[0079] (V) Torus and torus wedge

[0080] A closed surface formed by rotating a circle (⊙P) about an axis coplanar with that circle is called a torus (as shown in the figure below). The radius of ⊙P is called the outer radius, and the distance from the center of ⊙P to the center point of the torus is called the inner radius. The portion intercepted by the two planes passing through the z-axis is temporarily called the torus wedge (the correct name is unknown), and the angle between the two planes is called the axial angle. (The torus is shown below.) Figure 5-1 ) and toroidal wedges (such as Figure 5-2 ) Schematic diagram as follows Figure 5 As shown:

[0081] In a rectangular coordinate system, the equation of the torus symmetric about the z-axis azimuth angle is:

[0082]

[0083] Where R is the inner radius, r is the outer radius, and R>r>0.

[0084] II. Research and Development Principles

[0085] (I) Mathematical Formulas

[0086] The formulas for the perimeter (arc length), surface (arc area), and volume of the various shapes mentioned above in this application are shown in Table 1.

[0087] Table 1. List of Mathematical Formulas for Various Shapes:

[0088]

[0089] (II) Design Principles

[0090] The design principle of this achievement is to transform multiplication and division (including exponentiation) in the direct or indirect calculation process of various shape parameters into addition and subtraction operations, to transform addition and subtraction operations into "forward and reverse rotation of the toroidal angle", to solidify "various shape parameters and complex calculations" into "scale", and to transform "professional operations" into "simple rotation, alignment, and reading". Specific operations are described in subsequent sections.

[0091] (III) Algorithm Principle

[0092] The algorithm principles involved in this application mainly include the following aspects:

[0093] 1. Direct reading of isoparametric pairs

[0094] For calculations involving a set of univariate dependent variables with the same independent variable, the direct reading method using equivalent parameter comparison is applicable. A brief introduction is given below, using the quick calculation of a circle's circumference and area as an example (the formula is shown below):

[0095] (1) First, based on the arithmetic sequence of the common independent variable radius (r) of the circumference (L) and area (S), calculate the sequences of circumference (L) and area (S) respectively, and draw the following calculation table.

[0096] Table 2. Sample table for calculating the circumference (L) and area (S) of a circle

[0097] Radius (r) 0 1 2 3 4 5 … Perimeter (L) 0 6.28 12.56 18.84 25.12 31.40 … Area (S) 0 3.14 12.56 28.26 50.24 78.50 …

[0098] (2) Based on the principle of "equal dependent variables", scaled toroidals of r, L, and S are constructed (equivalent to bending Table 2 into an arc). In this way, the circumference (L) or area (S) can be directly read by looking up the radius (r) and comparing it with the L or S toroidal to achieve the purpose of quick calculation. The scaled toroidal torsion ... Figure 6 As shown:

[0099] 2. Indirect Operations

[0100] Indirect computation refers to a method of solving problems by exploring the relationships between a series of related variables or parameters and performing multiple calculations or transformations. It is often used in situations where direct computation is impossible or inconvenient. The process often involves multiple steps or combinations of multiple variables, requiring the construction of one or more intermediate computations related to the original computation, and then derivation and calculation are performed based on the relationships between these intermediate computations and the original computation.

[0101] To address the issue that multiplication and division operations cannot be directly converted to pairs, this project designed the following indirect calculation strategy:

[0102] (1) Taking the "logarithm": Multiplication and division operations become addition and subtraction operations.

[0103] By utilizing the properties of logarithmic functions, we can first take the logarithm of the multiplication and division expression, thus converting the multiplication and division operations into addition and subtraction operations, as shown in the following example:

[0104]

[0105] (2) Using angle algebra to calculate acceleration and deceleration

[0106] make Y a =log 10 a, Y b =log 10 b, Y c =log 10 c, then construct Y using the principle of "equal angles and equal values". a Y b Y c Four scaled rotating rings, Y0, and Y10, and connected through Y10 and Y20. a Y b Y c The quick calculation operation of the three scaled rotating rings, with the "first-to-last alignment angle added and the last-to-last alignment difference angle subtracted," yields Y. a +Y b -Y c Then Y a +Y b -Y c By aligning and comparing the Y0 rotation ring with its beginning and end, the Y0 value can be read directly, thus enabling acceleration and deceleration calculations through angle algebra.

[0107] 3. Equivalent Substitution Method

[0108] The equivalent substitution method, under the premise of "equal effect," replaces the original formula or parameter with a formula or parameter that has the same effect but is simpler or easier to calculate, thereby simplifying the calculation process or overcoming design difficulties. For example: when f(x) = F(x) and f(x) is simpler than F(x), f(x) can be used to replace F(x) for calculation; this is an equivalent substitution of the method. When f(x) = f(y,z) and the calculation is more convenient, x can be used to replace y and z for calculation; this is an equivalent substitution of the parameter. When f(x) = F(y,z) and the calculation is simpler and more convenient, f(x) can be used to replace F(y,z); this is an equivalent substitution of the method / parameter. In this work, all three types of equivalent substitution are used.

[0109] 4. Combination of multiple algorithms

[0110] Due to the complexity of calculating ellipsoidal shape parameters, it is impossible to solve all problems using a single quick calculation algorithm. Therefore, it is necessary to adhere to a problem-oriented approach, analyze specific problems individually, and design and comprehensively apply relevant quick calculation algorithms based on actual conditions to achieve the goal of rapid calculation of ellipsoidal shape parameters. Specific details will not be elaborated further.

[0111] III. Structure and Fabrication

[0112] (I) Structure

[0113] The elliptical abacus, with its derived geometric parameters, mainly consists of a rotating ring, markers, scales, pointers, and a fixed axis. Figure 7 As shown in Table 3. The rotating rings are connected by a connecting structure, allowing for flexible relative rotation. The ring markers indicate the meaning or function of the scale on the corresponding ring (see Table 3 for details). The scale markings are engraved on the rotating rings, enabling various calculation functions through the relative rotation between the rings. The pointer is fixed to the center of the abacus by a fixed shaft, serving as an auxiliary tool for scale alignment and reading.

[0114] Table 3. Overview of the meaning and functions of the ring table

[0115]

[0116]

[0117] (II) Production

[0118] 1. Material selection:

[0119] The rotating ring of the elliptical abacus can be made of cardboard, plastic board, wood board, metal, etc., the pointer can be made of metal wire or plastic strip, and the fixed axis can be made of metal.

[0120] 2. Making the rotating ring:

[0121] Use a knife or other tools to process the sheet material into rotating rings of equal size. The outer circumference of the ring should have a ring-shaped protrusion (except for the outer ring), and the inner circumference should have a ring-shaped groove (except for the center ring). The width, thickness, and position of the protrusion and the groove should be precisely consistent to ensure that the rotating rings can be perfectly combined and rotate flexibly.

[0122] 3. Calculation and Characterization:

[0123] Based on the aforementioned formulas for calculating the parameters of various elliptical derivative shapes, a series of scale values ​​are precisely calculated and accurately and clearly marked on the corresponding rings and positions.

[0124] 4. Control pointer:

[0125] The pointer is made of metal wire or plastic strip.

[0126] 5. Assembly:

[0127] The rings are inlaid together in sequence, and the pointer is fixed at the center of the elliptical system derived shape parameter abacus using a fixed axis.

[0128] 6. Packaging:

[0129] Use appropriate materials to make the packaging to enhance the protection of the elliptical-derived shape parameter abacus, so as to facilitate carrying and prevent deformation or damage.

[0130] IV. Operation

[0131] (I) Parameter Calculation of Circles and Spheres

[0132] The circumference and area of ​​a circle, and the surface area and volume of a sphere, can be calculated by directly looking up the values. Let's take the circumference of a circle as an example... Figure 8 As shown:

[0133] (1) Locate the r ring and ⊙L ring on surface A of the elliptical system derived shape parameter quick calculation disk and align their starting points (red line positions).

[0134] (2) Find the scale (point A) corresponding to the radius value of the circle in the r ring (regardless of the unit); if it is not an integer, use the interpolation method to estimate its position.

[0135] (3) Locate the scale mark (point B) on the circle L that corresponds to point A on the circle r. The value of this mark is the circumference of the circle (in the same unit as the radius). When the circle is between two scale marks, estimate its value using interpolation. If necessary, a pointer can be used.

[0136] The calculation of the area of ​​a circle, the surface area of ​​a sphere, and the volume of a sphere can be performed using the methods described above.

[0137] (II) Calculation of the circumference of the ellipse

[0138] A schematic diagram of the steps for calculating the circumference of an ellipse is shown below. Figure 9 As shown:

[0139] (1) According to the aforementioned method for calculating the circumference of a circle, find the circumference scale position (point B) of the circle with the minor semi-axis (b) of the ellipse as the radius (i.e., r = b) on the ⊙L ring.

[0140] (2) The starting point of the ring is aligned with point B of the ⊙L ring.

[0141] (3) In Find the mark (point C) on the ring corresponding to the difference between the major semi-axis (a) and the minor semi-axis (b) of the ellipse (i.e., ab).

[0142] (4) Find on the ⊙L ring The position corresponding to point C (point D) is the circumference of the ellipse.

[0143] (III) Calculation of the area of ​​the ellipse and the surface area of ​​the torus

[0144] A schematic diagram illustrating the steps for calculating the area of ​​an ellipse and the surface area of ​​a torus, as shown below. Figure 10 As shown, it involves two steps: calculating and verifying equivalent parameters.

[0145] (1) In ① Locate the position (point E) corresponding to the inner radius (R) of the torus on the torus; ② Locate the position (point F) corresponding to the outer radius (r) of the torus. Note: ① Ring and ② The meaning and function of the rings are exactly the same and they can be interchanged.

[0146] (2) ② The starting point of the ring is aligned ① Point E of the ring, at this time ② Point F of the ring is at ① The scale value of the corresponding point (point G) on the ring is the equivalent parameter (d) of the toroidal surface R*r.

[0147] (3) Find the position of the d value (point H) on the r ring, and the scale value of its corresponding position (point I) on the ⊙S ring is the area value of the ellipse.

[0148] The calculation method for the area of ​​an ellipse is the same as that for the surface area of ​​a torus, except that the inner radius (R) and outer radius (r) of the torus are replaced with the major semi-axis (a) and minor semi-axis (b) of the ellipse. Simply replace the ring with a ⊙S ring.

[0149] (iv) Volume calculation of ellipsoids and tori

[0150] like Figure 11 As shown, the calculation of torus volume and torus surface area is similar, and the calculation steps are as follows:

[0151] (1) First, find the toroidal surface area calculation method described above. ① The position (point G) on the ring corresponding to the equivalent parameter (d) of the toroidal surface R*r.

[0152] (2) Then ② The starting point of the ring is aligned ① Point G of the ring.

[0153] (3) In ② Find the position (point J) corresponding to the outer radius (r) on the ring. ① The scale value at the position (point K) corresponding to point J on the ring is the equivalent parameter (f) of R*r2.

[0154] (4) Locate the position (point L) corresponding to the value of f on the r-ring. The graduation on the ring corresponding to point M is the volume value of the torus.

[0155] The calculation methods for ellipsoidal volume and torus volume are the same; only the R*r*r of the torus needs to be replaced with the a*b*c of the ellipsoid. Simply replace the ring with a ●V ring.

[0156] (vi) Calculation of the surface area of ​​an ellipsoid

[0157] like Figure 12 , Figure 13 As shown, the method for calculating the surface area of ​​an ellipsoid is as follows:

[0158] (1) First, find the ellipsoidal volume using the aforementioned method. ① The position (point K) on the ring corresponding to the equivalent parameter (f) of the ellipsoid a*b*c.

[0159] (2) Locate the corresponding positions (point N and point P) of the f value and 2 / 3 of the f value on the r-ring, and align point N with point K. ① The scale value at the position corresponding to point P on the ring (point Q) is (abc). 2 / 3 The equivalent parameter (g).

[0160] (3) Find the position (point R) corresponding to the value of g on the r ring. At this time, the scale of the position (point S) corresponding to point R on the ●S ring is the surface area value of the ellipsoid.

[0161] (vii) Parameter calculation of sector, spherical wedge, ellipsoidal wedge and toroidal wedge

[0162] The calculations of sector arc length, sector area, spherical wedge sphere area, spherical wedge volume, ellipsoidal wedge sphere area, ellipsoidal wedge volume, torus wedge arc area, and torus wedge volume are based on the circumference of a circle, the surface area of ​​a sphere, the surface area of ​​an ellipsoid, and the surface area of ​​a torus, respectively. Only in the equivalent parameter step is it necessary to utilize... ③ The equivalent parameters of the ring can be further adjusted. For example... Figure 14 , 15 As shown, the calculation of the area of ​​a sector will be used as an example to illustrate the process:

[0163] (1) First in ③ Locate the corresponding central angle mark (point T) on the ring. Note: ③ Unlike other rings, this ring starts at 360° and its degrees are arranged in descending counterclockwise order.

[0164] (2) Align the starting point of ring r with point T, and... ③ Mark the starting point of the ring on the r-ring as point U, and find the position at 1 / 2U and mark it as point V.

[0165] (3) In ① Locate the sector radius value scale (point W) on the ring and align point W with it. ③ The starting point of the loop, at which point V is located ① The scale value at the corresponding position (point X) on the ring is the equivalent radius (h) of the sector.

[0166] (4) Find the corresponding position of h value (Y point) on the r ring. At this time, the scale value of the Y point on the corresponding position on the ⊙S ring is the area value of the sector.

[0167] The equivalent parameter adjustment methods for the remaining seven shape parameters are the same and can be followed accordingly. It is particularly important to note that: the equivalent parameter adjustment coefficient for the sector arc length (the ratio of arc UV / arc OU on the r-ring, see the figure above) is 1; the equivalent parameter adjustment coefficients for the sector area, spherical wedge sphere area, and toroidal wedge arc area are 1 / 2; the equivalent parameter adjustment coefficients for the spherical wedge volume, ellipsoidal wedge volume, and toroidal wedge volume are 1 / 3; and the equivalent parameter adjustment coefficient for the ellipsoidal wedge sphere area is 2 / 3.

[0168] This invention overcomes the shortcomings of manual calculation, which involves a large amount of labor and repetitive work. It simplifies the complex calculations and data searches of profound theories into a simple "one-to-one" operation. It has the advantages of ingenious conception, scientific design, simple structure, easy operation, easy promotion, strong applicability, low cost and portability, and has good prospects for promotion and application.

Claims

1. A quick abacus for elliptical-system derived shape parameters, comprising a central ring, characterized in that: The central ring is connected to the middle ring and the side rings in sequence by means of mortise and tenon joints; a pointer is fixed to the center of the central ring by a fixed shaft.

2. The elliptical system derived shape parameter quick abacus according to claim 1, characterized in that: The central ring, middle ring, and side ring are provided with scales.

3. The elliptical system derived shape parameter quick abacus according to claim 1, characterized in that: The pointer is rectangular with one side open, and one end of the opening is clamped to the center of the central ring through a fixed shaft.

4. The elliptical system derived shape parameter quick abacus according to claim 1, characterized in that: The central ring is a multi-ring structure with detachable mortise and tenon joints.

5. A method for performing calculations using a mental abacus with elliptical system derived shape parameters as described in claim 1, characterized in that... Includes the following steps: Parameter calculations for circles and spheres: The calculation methods for the circumference and area of ​​a circle, and the surface area and volume of a sphere, are directly based on the provided information. This includes the following steps, taking the circumference of a circle as an example: (1) Locate the r-ring and ⊙L-ring on surface A of the elliptical system derived shape parameter quick calculation disk and align their starting points (red line positions); (2) Find the scale point A corresponding to the radius value of the circle in the r ring; when it is not an integer, estimate its position using interpolation method; (3) Find the scale point B on the ⊙L ring that corresponds to point A on the r ring. The value of this scale point is the circumference of the circle, and the unit is the same as the radius unit. When it is between two scale points, use the interpolation method to estimate its value. The calculation of the area of ​​a circle, the surface area of ​​a sphere, and the volume of a sphere can be performed using the methods described above.

6. A method for performing calculations using a mental abacus with elliptical system derived shape parameters as described in claim 1, characterized in that... Includes the following steps: The steps for calculating the circumference of an ellipse are as follows: (1) According to the aforementioned method for calculating the circumference of a circle, find point B on the circle with radius b as the minor semi-axis of the ellipse on the ⊙L ring, i.e. r = b; (2) The starting point of the loop is aligned with point B of the ⊙L loop; (3) In Find the scale point C corresponding to the difference between the major semi-axis a and the minor semi-axis b of the ellipse on the ring, i.e., ab. (4) Find on the ⊙L ring Point D corresponds to point C on the ring; the scale value here represents the circumference of the ellipse.

7. A method for performing calculations using a mental abacus with elliptical system derived shape parameters as described in claim 1, characterized in that, Includes the following steps: The calculation of the area of ​​an ellipse and the surface area of ​​a torus involves two steps: equivalent parameter calculation and verification. Taking the calculation of the surface area of ​​a torus as an example: (1) In ① Locate point E on the ring, corresponding to the inner radius R of the ring; ② Locate point F, which corresponds to the outer radius r of the torus. ① Ring and ② The meaning and function of rings are exactly the same and they can be interchanged; (2) ② The starting point of the ring is aligned ① Point E of the ring, at this time ② Point F of the ring is at ① The scale value of the corresponding point G on the ring is the equivalent parameter d of the toroidal surface R*r; (3) Find the position H of the d value in the r ring, and the scale value of the corresponding position I on the ⊙S ring is the area value of the ellipse. The calculation method for the area of ​​an ellipse is the same as that for the surface area of ​​a torus, except that the inner radius R and outer radius r of the torus are replaced with the major semi-axis a and minor semi-axis b of the ellipse. Simply replace the ring with a ⊙S ring.

8. A method for performing calculations using a mental abacus with elliptical system derived shape parameters as described in claim 1, characterized in that... Includes the following steps: The calculation of torus volume is similar to that of torus surface area, and the calculation steps are as follows: (1) First, find the toroidal surface area calculation method described above. ① Point G on the ring, corresponding to the equivalent parameter (d) of the toroidal surface R*r; (2) Then ② The starting point of the ring is aligned ① Point G of the ring; (3) In ② Find point J on the ring corresponding to the outer radius r. At this time... ① The scale value of point K on the ring, which corresponds to point J, is the equivalent parameter f of R*r2; (4) Locate point L on the r-ring corresponding to the value of f. The scale mark at point M on the ring, corresponding to point M, represents the volume of the ring surface. The calculation methods for ellipsoidal volume and torus volume are the same; only the R*r*r of the torus needs to be replaced with the a*b*c of the ellipsoid. Simply replace the ring with a ●V ring.

9. A method for performing calculations using a mental abacus with elliptical system derived shape parameters as described in claim 1, characterized in that, Includes the following steps: The method for calculating the surface area of ​​an ellipsoid is as follows: (1) First, find the ellipsoidal volume using the aforementioned method. ① Point K on the ring, corresponding to the equivalent parameter f of the ellipsoid a*b*c; (2) Locate points N and P on the r-ring corresponding to the f value and 2 / 3 of the f value, and align point N with point K. At this point... ① The scale value at point Q on the ring, corresponding to point P, is (abc). 2 / 3 The equivalent parameter g; (3) Find the position R corresponding to the g value on the r ring. At this time, the scale of the position S on the ●S ring corresponding to the R point is the surface area value of the ellipsoid.

10. A method for performing calculations using a mental abacus with elliptical system derived shape parameters as described in claim 1, characterized in that... Includes the following steps: Parameter calculations for sector-shaped, spherical, ellipsoidal, and toroidal wedges: The calculations of sector arc length, sector area, spherical wedge sphere area, spherical wedge volume, ellipsoidal wedge sphere area, ellipsoidal wedge volume, torus wedge arc area, and torus wedge volume are based on the circumference of a circle, the surface area of ​​a sphere, the surface area of ​​an ellipsoid, and the surface area of ​​a torus, respectively. Only in the equivalent parameter step is it necessary to utilize... ③ Further adjustments to the equivalent parameters of the ring are needed; specifically, taking the calculation of the sector area as an example, the steps are as follows: (1) First in ③ Locate the corresponding central angle mark T on the ring; ③ Unlike other rings, the ring starts at 360° and the degrees are arranged in descending counterclockwise order; (2) Align the starting point of ring r with point T, and... ③ Mark the starting point of the ring on the r-ring as point U, and find the position at 1 / 2U and mark it as point V; (3) In ① Locate point W on the sector radius scale on the ring and align point W with the scale. ③ The starting point of the loop, at which point V is located ① The scale value at point X on the ring is the equivalent radius h of the sector; (4) Find the corresponding position Y of h value on the r ring. At this time, the scale value of the corresponding position of Y on the ⊙S ring is the area value of the sector. The equivalent parameter adjustment methods for the remaining 7 shape parameters are the same and should be performed in accordance with the above methods. It is important to note that: the equivalent parameter adjustment coefficient for the arc length of the sector (the ratio of arc UV / arc OU on the r ring, see the figure above) is 1; the equivalent parameter adjustment coefficients for the area of ​​the sector, the area of ​​the spherical wedge, and the area of ​​the toroidal wedge arc are 1 / 2; the equivalent parameter adjustment coefficients for the volume of the spherical wedge, the volume of the ellipsoidal wedge, and the volume of the toroidal wedge are 1 / 3; and the equivalent parameter adjustment coefficient for the area of ​​the ellipsoidal wedge is 2 / 3.

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