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Plotting the graph of a tilted elliptic function

To determine the direction of the major and minor axes of the ellipse, we can transform the equation to see if it is correct. We can eliminate the xy term by performing a linear transformation on x and y. First, we define new variables u and v as follows:
u = x + y
v = x - y
By plugging in these variables and doing some calculations, we can transform the original equation into this new one:
u^2 + 3 v ^2 = 4
This equation describes an ellipse, where the u and v axes correspond to the major and minor axes in the original equation. Since the coefficient of u^2 is greater than the coefficient of v^2, the u axis is the major axis of the ellipse and the v axis is the minor axis of the ellipse.
Therefore, the curve described by the original equation x^2 + y^2 + xy = 1 is a tilted ellipse whose major and minor axes are not parallel to the coordinate axes.

xy cross term introduces diagonal line

Relative to the standard circle, the function graph will shift to the left or to the right as a whole, and also follows the rule of "addition on the left and subtraction on the right".

Negative slash component

When we look at the graph of the equation “x^2 + y^2 + xy = 1” we notice that it is shifted to the left relative to the graph of the standard unit circle.
This is due to the existence of the xy cross term, which introduces an additional negative diagonal component into the image, causing the image to shift to the left as a whole.

Forward slash component

The function expression corresponding to the forward slash component can be expressed by decomposing the xy cross term. Suppose we use a new variable t = x + y to represent the sum of x and y. Then we can rewrite the equation "x^2 + y^2 + xy = 1" as:
(x + y)^2 - 2xy + xy = 1
Simplified:
t^2 - xy = 1
This new equation t^2 - xy = 1 describes the positive slope component. In this equation, x and y appear in the xy cross term, and the coefficient of this cross term is -1.
The function expression corresponding to the positive diagonal component is t^2 - xy = 1, where t = x + y. This equation describes the positive diagonal part of the image.

x^2 + y^2 + xy = 1 (negative)

clc,clear,close all;
% 定义方程
eqn = @(x, y) (x.^2 + y.^2 + x.* y ) - 1;

% 绘制方程曲线和坐标轴
ezplot(eqn, [-2, 2, -2, 2])
hold on  % 在同一图形中保持绘图

% 绘制 x 坐标轴
plot([-2, 2], [0, 0], 'k-')  % 绘制水平线段

% 绘制 y 坐标轴
plot([0, 0], [-2, 2], 'k-')  % 绘制垂直线段

hold off  % 结束绘图区域的保持

xlabel('y')
ylabel('x')
title('函数绘制结果')
grid on
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Plotting Results

x^2 + y^2 - xy = 1 (positive)

clc,clear,close all;
% 定义方程
eqn = @(x, y) (x.^2 + y.^2 - x.* y ) - 1;

% 绘制方程曲线和坐标轴
ezplot(eqn, [-2, 2, -2, 2])
hold on  % 在同一图形中保持绘图

% 绘制 x 坐标轴
plot([-2, 2], [0, 0], 'k-')  % 绘制水平线段

% 绘制 y 坐标轴
plot([0, 0], [-2, 2], 'k-')  % 绘制垂直线段

hold off  % 结束绘图区域的保持

xlabel('y')
ylabel('x')
title('函数绘制结果')
grid on
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Plotting Results