Ellipse covariance matrix. Draw Rotated Ellipses on Matlab.
Ellipse covariance matrix. Now consider the case where they they co-vary.
Ellipse covariance matrix P. Draw a two-dimensional ellipse that traces a bivariate normal density contour for a given mean vector, covariance matrix, and probability content. Now consider the case where they they co-vary. nsigma (int, optional): The number of standard deviations for the ellipse. The covariance matrix for the input data is calculated by the formula =COV(A4:B13), as shown in range D4:E5, with the inverse shown in range D11:E12 as calculated by =MINVERSE(D4:E5). </p> Sep 23, 2015 · If I have two eigenvalue $\\lambda_1$ and $\\lambda_2$ and two associated normalized eigenvector $\\mathbf e_1$ and $\\mathbf e_2$ respectively, and I want to draw ellipse, How can I know which eigenva 2 The covariance matrix The concept of the covariance matrix is vital to understanding multivariate Gaussian distributions. Apr 23, 2013 · Learn more about eigenvalues, covariance matrix . For an ellipse that is not centered on the standard coordinate system an example will show how to rotate the ellipse. 03), nrow = 2) ellipse(c(-0. cov(). You want to look at the eigenvalues of this inverse, not the diagonal elements. Note that σ Calculates ellipse parameters from covariance matrix. Figuring out probability ellipse from covariance matrix. T) eigenvalues, eigenvectors = np. Although GMM are often used for clustering, we can compare the obtained clusters with the actual classes from the dataset. The estimated mean vector of the combined distribution is then defined by: Apr 8, 2011 · The eigenvalues give the corresponding lengths of the major/minor axis of the ellipse, and so scaling the ellipse or the eigenvalues to the 95%tile is equivalent to multiplying the covariance matrix with the scaling factor. , (2, 2)) points = np. Then it can be shown, how to write the equation of an ellipse in terms of matrices. For example, this code creates an ellipse, whos major axis is the x-axis, and whos minor axis is the y-axis. method: the covariance method to be used: classical product-moment ("classical"), or minimum volume ellipsoid ("mve"), or minimum covariance determinant ("mcd"). $\endgroup$ – Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. npoints: Number of points comprising the border of the ellipse. g. 2. 59, 24. cova (np. The calculation for the covariance matrix can be also Jan 14, 2020 · where \(C_{\overline{x}}\) is the minimized covariance matrix of the combined probability distribution, and \(C^{-1}_{i}\) is the covariance matrix of the individual probability distributions. Thus, the volume (area) of the prediction ellipse is proportional to the square root of the generalized variance. Feb 20, 2011 · This brings us to our first “aha” moment: any correlation matrix is actually the covariance matrix of the z-values of the series, but the eigen values and vectors have no useful interpretation in the real world. For this, I first calculated the Covariance Matrix and its associated Eigenvalues: cov = np. With this method, one generally gets an ellipse aligned with the shape of the scatter plot. For instance, the first hit in the search gives code that starts right off with the covariance matrix. The radiuses of the ellipse can be controlled by n_std which is the number of standard deviations. , 10. 95%, major axis length, minor axis length, angle of rotation, and center of the ellipse. Apr 10, 2018 · By definition a covariance matrix is positive definite therefore all eigenvalues are positive and can be seen as a linear transformation to the data. Using the sample covariance matrix as an approximation for the population covariance matrix and the eigenvalues obtained in Figure 2, the lengths of these axes are: Figure 3 – Axes lengths for Example 1. One can show COVARIANCE, for example with an ISO LINE. Clarification about ellipsoidhull in R. Feb 2, 2018 · You can understand the shape of the ellipsoid better if you look at the spectral/eigen decomposition of the precision matrix (inverse of the covariance matrix). Calculate the elements of the standard confidence (error) ellipse of the above point. When covariance matrices are nearly equal, their covariance ellipses should all have the same shape. This function plots the confidence ellipse of the covariance of the given array-like variables x and y. data Feb 15, 2019 · Scaling the plot is equivalent to multiplying each point by a diagonal matrix D and the scaled covariance will thus be D C D (where C is the old covariance matrix). data The covariance ellipse represents an iso-contour of the Gaussian distribution and allows visualization of a \( 1\sigma \) confidence interval in two dimensions. In the open source stat program R, you can use the function ellipse. s. The CRLB Matrix Then, under the same kind of regularity conditions, the CRLB matrix is the inverse of the FIM: CRLB = I−1(θ) So what this means is: nn nn n [ ] [1()] ‹ 2 ‹ C I θ θ = ≥ − θ σ Diagonal elements of Inverse FIM bound the parameter variances, which are the diagonal elements of the parameter covariance matrix A function to draw the principal axes of a 2D ellipse from a correlation, covariance or sums of squares and cross products matrix in an existing plot. ax : The axis that the ellipse will be plotted on. 1. Returns: np. It is assumed that the largest eigenvalue is given first. com Sep 11, 2022 · In order to plot a confidence ellipse based on your covariance matrix, you need to identify the defining characteristics of a confidence ellipse: confidence interval e. distributed data is explained completely by its mean and its covariance matrix. It is also known as the variance-covariance matrix because the variance of each element is represented along the matrix’s major diagonal and the covariance is represented among the non-diagonal elements. V. My approach so far: The axis lengths correspond to the two eigenvalues of the covariance matrix and defining the "spread". Plotting the Covariance Ellipse¶ This notebook is duplicated from the repository linked to in this article. My aim is to determine, in a visual manner, if the two covariance matrices are significantly different in a statistical sense. When working with multiple variables, the covariance matrix provides a succinct Feb 10, 2019 · Eviews doesn't support that kind of graph ornamentation so I am going with javascript to draw the ellipse and make a front-end graph of sorts. 281] -0. , positive semi definitive and symmetric, will there be any new characteristics/intuition of determinant. confidence ellipse, for 2D normally distributed data. 0000 0. cov(X. I collected the necessary information and I would like to know how the covariance ellipse is drawn. I know that from the eigenvalues of the covariance matrix of the ellipse, the major and the minor axis of the ellipse can be computed. After assuming this and that about the sample ( distribution etc ), the main steps are as follows : Determine the covariance matrix of the sample. linalg Aug 30, 2022 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have The covariance matrix gives the shape of how data is distributed in the feature space. 725 ]cm. Here is the scatter plot I am working with: Here is a list of (hopefully) all relevant descriptive stats that I was able to scrape from Eviews: I have a location of landmark in 2D. Expects a 2-element sequence of [x0, y0]. The ellipse is centered at the point representing the sample mean values of the two variables. This is a function of the square root of the generalized variance (given by the square root of the determinant of the variance-covariance matrix). Calculate the elements of the 95% confidence (error) ellipse of the above point. is along the ellipse’s major axis, the correlation matrix is σ′ = σ′2 1 0 0 σ′2 2 . uniform(0. Unfortunately, covariance matrices are often skipped over Covariance[a, b] gives the cross-covariance matrix for the matrices a and b. See full list on cookierobotics. Follow 0. Demonstration of several covariances types for Gaussian mixture models. EDIT. The optional argument ‘tol’ sets the tolerance for the verification of the positive-(semi)definiteness of the matrix K (see isdefinite). Note that when plotting confidence ellipses for data, the ellipse-axes are usually scaled to have length = square-root of the corresponding eigenvalues, and this is what the Cholesky decomposition gives. d. k. Skip to contents heplots 1. Mar 6, 2017 · Let: m = (meanX) (meanY) be the (vector) mean and: S = (Co00 Co01) (Co01 Co11) be the covariance matrix of the data. This can be a classical covariance matrix or a robust covariance matrix. Symmetric Matrices The symmetric matrix is a matrix in which the numbers on ether side of the This will be the center of the prediction ellipse. Cheng, although you might already know this, I suggest that you also read this answer to a question on randomness The matrix R is called the sample correlation matrix for the original data matrix X. According to Extended Kalman Filter EKF- SLAM, if the robot re-observes the same landmark, the covariance ellipse will shrink. 5 plotting the eigenvector of covariance matrix using matplotlib and np. multivariate_normal(mean=(0, 0), cov=cov, size=1000) # The 2x2 covariance matrix to base the ellipse on. The default value is alpha = . First thing you should do is find the covariance matrix using the method numpy. After you’ve found the covariance matrix, you can use the method numpy. Covariance[a] gives the auto-covariance matrix for observations in matrix a. ndarray): The center of the ellipse. If only one output argument is supplied a vector defining a ellipse is returned as defined in ellipses2d. 0 (0) 287 Downloads. 0 License. The actual radii of the ellipse are \(\sqrt{\lambda_1}\) and \(\sqrt{\lambda_2}\) for the two eigenvalues \({\lambda_1}\) and \({\lambda_2}\) of the scaled covariance matrix \(s\cdot\Sigma\). My confidence ellipse is described by the length of both axis and the angle of ellipse rotation. rotation) where the the off-diagonal terms of the matrix are zero. Apr 11, 2020 · For X = (X1, X2, X3) distributed as N3(µ, Σ), mean of the original data is mu and variance-covarinace matrix of the original data is Sigma. For full control over the plot and the analysis, it is useful to know how to do the calculations yourself. The volume of the 95% confidence ellipse is 24,842,086 calculated as follows: Figure 4 – Volume of 95% confidence ellipse for Example 1 2) Construct the covariance matrix: ? K R I = P L F : ê T ; 6 ê T U ê T U : ê U ; 6 G 3) Calculate the lengths of the ellipse axes, which are the square root of the eigenvalues of the covariance matrix: A E C R = H L A E C A J R = H Q A O : ? ; 4) Calculate the counter‐clockwise rotation (θ) of the ellipse: à L Note: we would call the matrix symmetric if the elements \(a^{ij}\) are equal to \(a^{ji}\) for each i and j. For example, given the matrix X defined as X = x1 x2 (3) The covariance of matrix X is COV(X) = σ2 x1 σ x1,x2 σ x1,x2 σ 2 x2 (4) The diagonal elements of a covariance matrix are the variances of the individual variables. All it needs is the coordinates of the center of the ellipse and the variance-covariance matrix of data XY (variances on the An ellipsoid centered at the origin is defined by the solutions $\mathbf{x}$ to the equation $\mathbf{x}^TM\mathbf{x} = 1$, where M is a positive definite matrix. The non-diagonal elements are covariances. Oct 28, 2016 · By following this post one can draw an ellipse with a given shape matrix (A): library(car) A <- matrix(c(20. In a previous article about or, more generally for correlated data with a covariance matrix having off-diagonal elements, with the covariance matrix C and a scaling parameter s (s is the square of sigma from above). Value A matrix with columns x and y to outline the Mar 21, 2023 · Next, we will plot the ellipse or the border of the ellipsoid. npoints (int, optional): The number of points to generate on the ellipse. Covariance Dec 10, 2014 · The ellipse code proposed in the original question is wrong. The estimated coordinates of a 0. , 95%) and the covariance matrix. linalg. RMIT University Geospatial Science N N N Ellipse Dec 12, 2022 · The problem is that definitions I've come across differ in how they end up calculating the actual area of this ellipse. Since this is a 6x6 covariance matrix, the odds that that diagonalizing transformation has any physical meaning is minimal. Figure 3 illustrates how the overall shape of the data defines the covariance matrix: If you want the ellipse to represent a specific level of standard deviation, the correct way of doing is by scaling the covariance matrix: STD = 2; %# 2 standard deviations conf = 2*normcdf(STD)-1; %# covers around 95% of population scale = chi2inv(conf,2); %# inverse chi-squared with dof=#dimensions Cov = cov(X0) * scale; [V D] = eig(Cov); Aug 3, 2018 · Also the covariance matrix is symmetric since \(\sigma(x_i, x_j) = \sigma(x_j, x_i)\). $\endgroup$ – a column matrix giving the mean of the two dimensions of the p-dimensional multivariate normal distribution. In this case its a $2\times 2$ matrix Nov 17, 2016 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Question: station U has a covariance matrix Cx = . The figure indicates three different classes and the red line indicates the same Mahalanobis distance for each class. I fitted my data with some parameters and now I have their optimal values popt and their covariance matrix pcov. Figure 1 – Confidence ellipse (part 1) Set-up. Taking the eigenvectors finds a space (i. IMHO, plotting ellipse only requires a $2 \times 2$ covariance matrix. However, it's simpler to use the Cholesky decomposition. I assume that these uncertainties are Gaussian and I would like to have a plot of er Sep 14, 2018 · An elegant and exact way to plot the confidence ellipse of a covariance. The ellipse is plotted into the given Axes object ax. If missing, a single covariance ellipse is drawn. cov = np. By using correlation rather than covariance we have given up some useful information: the actual variances! Aug 7, 2018 · """ # Generate some random, correlated data cov = np. How to draw a covariance error ellipse? In this post, I will show how to draw an error ellipse, a. In contrast the slopes of the lines will only be scaled by D . The scaling parameter s is chi-square distributed (sum of two Gaussian variables). 281 2. To plot the ellipse using matplotlib, we use the parameter transformation. That said, there is always a transformation that will make any nonsingular covariance matrix become diagonal. The form of the covariance matrix σ in the unrotated system follows from equation (14) using R Figure 1. Any ellipse can be described by four parameters: Ellipse center \( \mu_{x}, \mu_{y} \) Sep 18, 2024 · A Covariance Matrix is a type of matrix used to describe the covariance values between two items in a random vector. n: The number of nonmissing observations in the sample. Sep 13, 2024 · Here’s how the correlation value affects the covariance ellipse: Step 2: Covariance Matrix—After standardizing, the next step is to compute the dataset's covariance matrix. A covariance matrix gives the covariances of a set of variables. O. Confidence Ellipse. 43, -8. This should be of the same form as the output of eigen, namely a list with two components: values and vectors. 09), shape=A, radius=1. From G. eig: the eigenvalues and eigenvectors of the covariance matrix. eig(M) to find eigenvectors and Aug 24, 2017 · Each data matrix yields a covariance matrix using the standard sample covariance estimate. cov(points, rowvar=False) # Location of the center of the ellipse. newplot: If newplot = TRUE and draw = TRUE, plot the ellipse on a new plot. Apr 25, 2019 · TL;DR: Just add zeros. Jan 18, 2013 · Statistics 101: The Covariance MatrixIn this video, we discuss the anatomy of a covariance matrix. */ 0}; sigma={1. For this reason, the covariance matrix is sometimes called the _variance-covariance matrix_. p: The confidence level for the prediction ellipse Jul 23, 2015 · % 'C' - Alternate method of specifying the covariance matrix % 'mu' - Alternate method of specifying the ellipse (-oid) center % 'conf' - A value betwen 0 and 1 specifying the confidence interval. Recall that for a pair of random variables X and Y, their covariance is defined as Cov[X,Y] = E[(X −E[X])(Y −E[Y])] = E[XY]−E[X]E[Y]. e. a. How to apply rotation to an ellipse Feb 26, 2019 · You can extract them using the code in this gist, but you can also just run the eigen-decomposition of the given covariance-matrix. 0. 2D confidence ellipse for normally distributed data These directions are actually the directions in which the data varies the most, and are defined by the covariance matrix. Defaults to 1. 59,-8. 05, which results in a 95% ellipse. */ e=eigvec(sigma Aug 3, 2018 · I think that you might be overlooking the fact that the matrix form of the quadric equation isn’t always $\mathbf x^TQ\mathbf x=0$. In a previous article about Package ‘ellipse’ The summary function is used to obtain the covariance matrix of the fitted parameters. While working solutions where available, I had the idea that there should be a simpler and more elegant way. Its size and shape are determined by the chosen confidence level (e. This is what 3 covariance matrices look like Sep 6, 2012 · Parameters ----- cov : The 2x2 covariance matrix to base the ellipse on pos : The location of the center of the ellipse. b. 4 GMM covariances#. Updated 10 Nov 2016 Aug 5, 2015 · In this link, the ellipse of a covariance matrix is discussed in more details. The center of the ellipse is m, the axes of the ellipse are the eigenvectors e1 and e2 of the matrix S (the minor axis e1 is associated to the smallest eigenvalue of S). Code, explanation, examples and proof. Q zz = R Q xx R T where and Expand Q zz yields Apr 26, 2010 · Shows how the Gaussian error ellipse changes its shape as a function of the covariance matrix, the eigenvalues, and the correlation coefficient. Now, if we narrow down the definition of the matrix here: assuming that it is a covariance matrix, i. In this article, we provide an intuitive, geometric interpretation of the covariance matrix, by exploring the relation between linear transformations and the resulting data covariance. (25) Here, σ′ 1 is the 1-sigma confidence value along the minor axis of the ellipse, and σ′ 2 is that along the major axis (σ′ 2 ≥ σ′ 1). Mar 5, 2011 · It decomposes the matrix into 3 principal vectors using PCA, and visualizes them (+/- 3 sigma width) using rviz markers. Apr 28, 2015 · The easiest way to generate an ellipsoid with the right location and alignment given your distribution is to feed the mean and covariance directly to the Ellipsoid function, simply as Ellipsoid[mean, cov]. May 27, 2021 · Given a 2-dimensional dataset, I would like to plot an Ellipse around the data. 1 Draw ellipse based on data. See Gaussian mixture models for more information on the estimator. Thanks Jun 8, 2022 · I am trying to replicate the plot demonstrated in this guide, where a confidence ellipse is shown on top of a scatter plot, along with the eigen vectors. An ellipse can be described by the following equation: (x^2 / a^2) + (y^2 / b^2) = 1, where a and b are the lengths of the semi-axes of the ellipsoid. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. $$ The coefficients can be packaged up into a single homogeneous matrix $\mathtt Q$ by setting $\hat{\mathbf x}=(\mathbf x^T; 1)^T$ so that the equation becomes $$\hat{\mathbf x}^T\mathtt Q\hat{\mathbf x Draws a (1- alpha )100% confidence ellipse (two dimensional) for a multivariate normal distribution using the eigendecomposition of the covariance matrix. */ 0. Apr 29, 2014 · Covariance Matrix of an Ellipse. pooled: Logical; if TRUE, the pooled covariance matrix for the total sample is also computed and plotted. nstd : The radius of the ellipse in numbers of standard deviations. Dec 31, 2020 · As explained in intuition of determinant: we know that determinant is the how the volume is scaled when the matrix is regarded as a projection. 44, col= The covariance matrix represents the dispersion of data points while the inverse of the covariance matrix represents the tightness of data points. How is the dispersion and tightness related geometrically? For example, the determinant of the covariance matrix represents the volume of the dispersion of data points. After scavenging the internet for a Python Nov 10, 2016 · Visualise a 2x2 covariance matrix by plotting ellipses at 1, 2 and 3 standard deviations. Covariance[dist] gives the auto-covariance matrix for the multivariate symbolic distribution dist. Basically, the major axis can be determined by finding the largest eigenvalue of the covariance matrix and the the chi square value with given confidence level, thus $$ \text{L} = \sqrt{(\chi^{2}_{n,\alpha}) \lambda_{max}} $$ where L is half of the major axis length. random. Explore math with our beautiful, free online graphing calculator. . Mar 9, 2022 · I'd like to derive the covariance matrix that defines a given ellipse. 05, 0. Feb 5, 2019 · Covariance Matrix of an Ellipse. The covariance matrix can be considered as a matrix that linearly transformed some original data to obtain the currently observed data. n Figure 1. Similarly, a covariance matrix is used to capture the spread of three-dimensional data, and a covariance matrix captures the spread of N-dimensional data. An ellipse angle of 0 means, there's no correlation between x & y. The eigen_vectors_ give you the direction of the ellipses' axes and the corresponding eigen_values_ give you the square of the length along the eigenvector where the vertex is located. Usually, \(\textbf{A}\) is taken to be either the variance-covariance matrix \(Σ\), the correlation matrix, or their estimates S and R, respectively. 1 How to plot ellipse. Plot the standard ellipse in a reasonable scale. L. 494 -0. ndarray: The coordinates of the ellipse. Covariance[dist, i, j] gives the (i, j)\[Null]^th covariance for the multivariate symbolic Feb 10, 2022 · Convert position confidence ellipse to covariance matrix. The diagonal entries of the covariance matrix are the variances and the other entries are the covariances. 5000 2. Feb 4, 2020 · $\begingroup$ It would be difficult to draw such an ellipse without using the covariance matrix in the first place! Every answer goes through that process, even if it begins with a dataset. As the following: a1 = 2*sqrt(e1) a2 = 2*sqrt(e2) where a1 and a2 are the major and minor axis, e1 and e2 are the eigenvalues of covariance matrix. C. Derivation of how we can visualize high dimensional Gaussians conceptually using ellipses of equal probability in 2D, with examples for uncorrelated covarian May 29, 2024 · If missing, a single covariance ellipse is drawn. a. Defaults to 1000. A 2x2 matrix giving the covariance matrix. These plot methods provide one way to visualize possible heterogeneity of within-group covariance matrices in a one-way MANOVA design. Do you know how I can do this with MATLAB or any other method. Get the full Provisions are provided to do this for more than two variables, in a scatterplot matrix format. considered. cpp. The form of the covariance matrix σ in the unrotated system follows from equation (14) using R Jan 11, 2022 · There are correlations in the perifocal frame; I mentioned some in my answer. They include: Finding the contour level corresponding to the desired confidence (without this the result scarcely can be called a "confidence" ellipse!); Args: center (np. K must be symmetric positive (semi)definite. 5 Mahalanobis distance and mean-centered ellipse Sample covariance is p. If you have $\mathbf x = [x,y,z]^T$, then the zero vector satisfies this equation, so it can only represent quadrics that pass through the origin. 7. It's assumed that the matrix also has a 3x1 vector associated with it, which sets the mean x-y-z position. Most textbooks explain the shape of data based on the concept of covariance matrices. The general quadric equation can be broken down into the sum of a quadratic form, a linear term and a constant: $$\mathbf x^T\mathtt A\mathbf x+2\mathbf b^T\mathbf x+c=0. Draw Rotated Ellipses on Matlab. 0000}; lambda=eigval(sigma); /*The statements below calculate the xy coordinates for plotting the ellipse from the polar coordinates that are provided above. The covariance ellipse provides a geometric interpretation of the covariance matrix. how to draw an ellipse with given equation in Matlab? 1. I found in this section that we can derive the variance-covariance matrix of the Principal component using eigen vectors. eig(cov) I would now like to plot an Ellipse around the two Eigenvectors using matplotlib, but can't figure out, how. ndarray): The covariance matrix. Apr 3, 2014 · In this post, I will show how to draw an error ellipse, a. The following figure shows a 95% confidence ellipse for a set of 2D normally distributed data samples. All points lying on the red line have the same distance from the class mean, because it is used the covariance matrix. Years ago, I was looking for a recipe to plot the confidence ellipse of a covariance. Confidence ellipses are extension of the concept of a confidence interval, which is used for a single variable. Apr 19, 2023 · How do you find eigenvalues and eigenvectors from the covariance matrix? You can find both eigenvectors and eigenvalues using NumPy in Python. alpha: Probability to be excluded from the ellipse. Eigenvalues and eigenvectors are used for: Computing prediction and confidence ellipses COMPUTATION OF ELLIPSE AXIS Problem is to develop a new covariance matrix from existing Q xx matrix which removes correlation between unknown coordinates. S: A covariance matrix. Covariance matrix without correlation. The code is here, relevant function is at line 218: feature_viewer. The error ellipse represents an iso-contour of the Gaussian distribution, and allows you to visualize a 2D confidence interval. Information I have: length of major axis $\lambda_1$ length of minor axis $\lambda_2$ angle of rotation of the ellipse is $\theta$ degrees; determinant of the desired covariance matrix is $ = 1$ Aug 28, 2014 · In that case, the eigenvalues of the covariance matrix are the diagonal of the covariance matrix. An Alternative Way to Plot the Covariance Ellipse by Carsten Schelp, which has a GPL-3. Jul 23, 2014 · You can compute a prediction ellipse for sample data if you provide the following information: m: A vector for the center of the ellipse. Defaults to 2 standard deviations. Recall that the sample covariance is S = 1 n 1 Xn i=1 (~x i ~x )(~x i ~x )>: Is S always positive semide nite? Consider the spectral decomposition S = Xp j=1 j~u j~u > j: Then S~u j Aug 9, 2014 · I have a 3D scatter plot and I want to visually show COVARIANCE on it. You could extract the eigenvectors and -values via eigen(A). It works when the X1 and X2 variables have a mean of 0 and a standard deviation of 1, but not in the general case. The geometrical objects that come to mind are ellipses associated with the covariance matrices. 5000, /*This specifies the values of the covariance matrix, which must be symmetric. The location of a landmark is $<\!x:30,y:60\!>$. Nov 2, 2020 · $\begingroup$ @whuber From your answer, I feel that you plotted an ellipse, not an ellipsoid. efmgaozetlqrnwprrksonxbuigtuooshouijunrlcdkkbbl