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We have discussed n leaf knots. When n is greater, you will find that the image is more like a ring. This section describes several other curves wound around the ring. vertices = 12000t = from 0 to (64*PI)p = rand_int2(2, 32)q = rand_int2(2, 32)r = 2 + cos(q/p*t)x = r*sin(t)y = sin(q/p*t)z = r*cos(t)r = 0.5 + 0.5*sin(t)g = 0.5 + 0.5*yb = 0.5 + 0.5*cos(t) Curve

is called {O; I, the transformation from j} to {O '; I', j '} isRotating Coordinate Transformation. Rotation and Transformation Formula Since round (I, I ') = 0, round (I, J') = + θ Then I '= cos θ I + sin θ j, J' = cos (+ θ) I + sin (+ θ) j =-sin θ I + cos θ j ∴ Xi + yj = x 'I' + y 'J' = x' (cos θ I + sin θ j) + y' (

method in the second column (B, E, H) of the second matrixThe result of the second column is displayed.Repeat the above action in the second row to get the result of the second row:If the number of rows in the first matrix is greater than 1,U V WA B CX y z * D E FG h IThe 3 × 2 matrix is obtained:(U * A + V * D + W * g) (u * B + V * E + W * H) (u * C + V * F + W * I)(X * A + y * D + z * g) (x * B + y * E + z * H) (x * C + y * F + z * I)Now let's take a look at some of the actually used matrix m

Common two-dimensional graphics commands: Plot: Plotting two-dimensional graphics Loglog: plotting with full logarithmic coordinates SEMILOGX: plotting with semi-logarithmic coordinates (X) Semilogy: Drawing with semi-logarithmic coordinates (Y) Fill: Draw Two-dimensional, multi-filled graphics polar: drawing Polar Chart ba R: Draw a bar graph stem: Draw a discrete sequence data graph stairs: Draw a ladder diagram ErrorBar: Draw the Error bar chart hist: Picture histogram Fplot: Draw function Di

it. Other cases can be proved similarly.Using the inverse proof method, we assume that all the edges of the smallest rectangle that covers the convex hull are not overlapped with the edges of the convex hull. That is to say, each edge of the smallest rectangle has only one vertex of the convex hull. As shown in 3, the rectangle ABCD is the smallest rectangle covering the convex hull. M, N, P, and q are the vertices of the convex hull on the four sides of the rectangle. Let's create mm 'CD, NN '

1. Set constant $a_{1},a_{2},\cdots,a_{n}$ meet $a_{1}+a_{2}+\cdots+a_{n}=0$, verify:$$\lim_{x\to \infty}\sum_{k=1}^{n}a_{k}\sin \sqrt{x+k}$$Proof. First, the conclusion of easy evidence$$\lim_{x \to \infty}\sin \sqrt{x+k}-\sin \sqrt{x+n}=0$$_{n}=-a_{1}-a_{2}-\cdots-a_{n-1}$ the $a into $\sum_{k=1}^{n}a_{k}\sin \sqrt{x+k}$.$$\lim_{x\to \infty}\sum_{k=1}^{n}a_{k}\sin \sqrt{x+k}=\lim_{x\to \infty}\sum_{k=1}^{n-1}a_{k} (\sin \sqrt{x+k}-\ Sin \sqrt{x+n}) =0$$The certificate is completed.2. If you ha

, although the latitude is converted to a distance multiplied by a constant, the Longitude Distance is calculated using the trigonometric function. the formula is as follows: R = earth’s radiusΔlat = lat2 lat1Δlng = lng2 lng1a = sin(Δlat/2) + cos(lat1) * cos(lat2) * sin(Δlng/2)c = 2*atan2(√a, √(1a))dist = R*c Compile an SQL query statement based on the formula: mysql> set @er=6366.564864;#earth’s radius (km

the average of the equivalent RGB VA Lues.The HSI definition of saturation is a measure of a color ' s purity/grayness. Purer colors has a saturation value closer to 1, while grayer colors has a saturation value closer to 0. (In other color models, the meanings and mathematical definitions of "saturation" is slightly different. See HSL and HSV color models for comparison.)Equations to Convert RGB values to HSI valuesSuppose R, G, and B are the red, green, and blue values of a color. The HSI int

, and the average radius is 6371.004 km. If we assume that the Earth is a perfect sphere, its radius is the mean radius of the Earth, which is recorded as R. If the zero-degree longitude line is used as the basis, the distance between the two points can be calculated based on the latitude and longitude of any two points on the Earth's surface. (here, the error caused by the calculation of the earth's surface topography is ignored, only theoretical estimates ). Set the longitude and latitude of t

Spiral Pipe of mathematical graphicsFor related software, see: Mathematical graphics visualization tool. Use script code with custom syntax to generate mathematical graphics. (1) Spiral Tube copy code vertices = D1: 720 D2: 72u = from 0 to (10 * PI) D1v = from 0 to (2 * PI) d2 a = 3 x = (a + cos (v) * cos (u) z = (a + cos (v) * sin (u) y = sin (v) + u copy code (

following formula is obtained for calculating the distance between two points: C = sin (MLatA) * sin (MLatB) * cos (MLonA-MLonB) + cos (MLatA) * cos (MLatB) Distance = R * Arccos (C) * Pi/180 Here, the units of R and Distance are the same. If 6371.004 km is used as the radius, Distance is the unit of kilometer. If other units such as mile are used, unit conversi

(Image source: "Astronomical Algorithm")What is the "parallax angle"? The parallax angle is the angle between the equatorial meridian of the celestial body and the plane meridian of the celestial body, before the Zhongtian is negative and the transit is positive. For first quarter, the parallax angle is the degree to which it "falls":(Image source: "Astronomical Algorithm")So how do we calculate the angle? The Book of astronomical algorithms gives a formula:\ (\tan{q} = \frac{\sin{t}}{\tan{\varp

performed on custom controls that expose messages. In general, transform is the implementation of matrix, for matrix, let's first make a question: It is known that the center of the circle O (0, 0) has a p (x, y) on the coordinate axis, and the op a degree is rotated counter-clockwise so that the point P reaches p1 (x1, Y1), with X, Y indicates the coordinates of Point P1. Solution: it is clear that P1 o is equal to Po, which is an arbitrary M point on the X axis. Assume that our mo

method, the final deduced formula A is: $s = acos(cos($radLat1)*cos($radLat2)*cos($radLng1-$radLng2)+sin($radLat1)*sin($radLat2))*$R; At present, most of the online use of Google public Distance computing company, the derivation Formula B is: $s = 2*asin(sqrt(pow(sin(($radLat1-$radLat2)/2),2)+

radius is 6356.755 km, and the average radius is 6371.004 km. If we assume that the Earth is a perfect sphere, its radius is the mean radius of the Earth, which is recorded as R. If the zero-degree longitude line is used as the basis, the distance between the two points can be calculated based on the latitude and longitude of any two points on the Earth's surface. (here, the error caused by the calculation of the earth's surface topography is ignored, only theoretical estimates ). Set the longi

* Math. sin (x3)-(x6 + 2 * x4) * Math. cos (x3) + Math. Exp (x0) * (x5 * Math. sin (x2)-(2 * x6 + x4) * Math. cos (x2 )));N_p [2] = (2* Math. exp (x0 + x1)* (X4 + x6) * Math. cos (x3) * Math. cos (x2)-x5 * Math. cos (x3)* Math. sin (x2)-x7 * Math.

monotonicity of the vector, whether the spacing and whether there are repeated elements.Diff)Syms x TF = cos (A * X)DF = diff (f) is defined by the findsym rule and is implicitly specified to differentiate X.DFA = diff (F, 'A') specifies the differentiation of variableDFA = diff (F, 'A', 3) triplicateThe diff function not only applies to the scalar, but also to the matrix. The calculation rule is to perform the differential operation based on the ele

integral is 0"Figure VI: FlowchartWhen you get here, you may have two states:One is: Ah, the original is so, I understand.One is: Ah, how can this, I completely can not imagine. The point here is that you don't have to imagine (visualize). The mathematics is so defined orthogonal, the mathematical proof of their orthogonality, then they are orthogonal, "they can not interfere with each other to carry their own information." Choose sin (t) and sin (2t) as an example, and formally because they ar

understand the four-dollar number, you cannot understand the code. The following article gives us a general understanding of what a four-dollar number is. We're not studying math, so we don't have to delve into it, as long as it helps us understand the code.In the article, there is this fragment:Given a Euler rotation (x, y, Z) (that is, rotate X, y, and z on the x-axis, y-axis, and z-axis, respectively), the corresponding four-dollar number isx = sin (y/2) sin (z/2)

the dry differential this inherent difficulty. So avoid numerical differentiation as much as possible. In particular, the data obtained from the experiment is differentiated. In this case, it is best to use the least squares curve to fit this data, and then differential the resulting polynomial, or a different method of point data three times spline fitting, and then look for spline differential, but sometimes the differential operation is unavoidable, in MATLAB. Use the function diff juice to