Understanding the Ellipse Equation | Maths Explanation for C++ Kids
In geometry, an ellipse can be defined as the set of all points where the sum of the distances to two foci is constant. The general form of an ellipse equation is (x - h)² / a² + (y - k)² / b² = 1. We'll use this to calculate ellipse points dynamically in C++. In this tutorial, we'll explore the ellipse equation in C++, derive its conic form, and show how to draw and animate ellipses on an C++ window frame using simple C++ code.
Using the Equation of an Ellipse in C++
Ellipses are represented by the general equation
| (x - h)2 | + | (y - k)2 | = 1 |
| a2 | b2 |
To animate motion along the ellipse in C++, we'll express *y* in terms of *x*.
Solving for y, we have:
| (y - k)2 | = 1 - | (x - h)2 |
| b2 | a2 | |
| (y - k)2 | = | a2 - (x - h)2 |
| b2 | a2 |
| (y - k)2 = b2[ | a2 - (x - h)2 | ] |
| a2 | ||
| (y - k) = | ±b√(a2 - (x - h)2) | |
| a |
y = k ± b/a√(a2 - (x - h)2)
This gives us two values for *y* (the upper and lower halves of the ellipse). We can use these values to plot and animate a moving object in C++ on both curves.
Parametric Equations of a Ellipse | Maths Explanation for C++ Kids
Elliptical motion can also be represented parametrically:
y = k + b * sin(θ)
These equations allow you to move an object smoothly along an elliptical trajectory on the canvas in C++, especially for animations, games, and physics simulations.
Animating Elliptical Motion with C++
To move a small image body (or dot) along an elliptical trajectory, we'll increment *x* from *(h - a)* to *(h + a)* and compute *y* using the ellipse equation in C++.
Create a new C++ project;
call it Dymetric.
Create 2 new C++ class files;
Call them Facet and EllipticalPath.
Type out the adjoining C++ code for animating an image body through the path of an ellipse.
How the C++ Ellipse Equation Animation Code Works
When you click the Move button, a dot will trace the upper and lower halves of the ellipse, showing how the ellipse equation in C++ generates smooth, curved motion.
Key Takeaways on Elliptical Path Animation in C++
In this tutorial, you've learned that:
- The general form of the ellipse equation is (x - h)² / a² + (y - k)² / b² = 1
- With an ellipse equation, you can draw and animate an ellipse in C++
- Applications of elliptical path animation include use in graphics, game design, and physics simulations
Applications of Ellipse Equations
Elliptical motion is used in physics simulations, orbital paths, and data visualization. By understanding the ellipse equation in C++, you can create animations, model celestial orbits, or generate smooth transitions in web graphics.
FAQs: Ellipse Equation and C++
What is the equation of an ellipse in C++?
The equation (x - h)² / a² + (y - k)² / b² = 1 can be implemented using C++ to compute ellipse points or draw an ellipse on a canvas element.
How else do you animate an ellipse path on canvas?
Use the parametric form x = h + a * cos(t), y = k + b * sin(t), and increment θ over time using
requestAnimationFrame() to create smooth elliptical motion.
Can you draw ellipses using the Canvas API directly?
Yes. The ctx.ellipse() method in modern browsers allows you to draw an ellipse directly without manually computing each point.
Summary: Visualizing ellipse Equation in C++
Ellipses are fundamental conic sections, and in C++, you can use their equations to draw and animate motion along an elliptical path on an C++ window frame.
In this lesson, we've derived the ellipse equation, express it in code, and use C++ to animate an image body moving through an ellipse.
So! C++ Fun Practice Exercise - Animate along Elliptical Path
As a fun practice exercise, try adjusting the foci points - a, b;
and the major and minor radii - h, k; to change the ellipse's position and size.
This will be a great way to connect mathematics and programming, and help you
understand more about C++ animations and ellipse equations.
C++ Elliptical Path Dymetric Window Code Stub
C++ Elliptical Path Canvas Frame and Button Controls - Header File
class Facet
{
public:
Facet(HWND, int, int);
virtual ~Facet();
bool decorateButton(WPARAM, LPARAM);
bool actionPerformed(HWND, WPARAM, LPARAM);
void informPaint();
};
C++ Elliptical Path Canvas Frame and Button Controls - Class File
#include "Facet.h"
#include "EllipticalPath.h"
EllipticalPath* elp_path;
/*
* Our custom class that interfaces between the parent window
* and the subsequent daemonstrator classes
*/
Facet::Facet(HWND hWnd, int window_width, int window_height)
{
elp_path = new EllipticalPath(hWnd, window_width, window_height);
}
/*
* This guy decorates buttons with colour and title text
*/
bool Facet::decorateButton(WPARAM wParam, LPARAM lParam) {
// button glide calling
if (wParam == 12321)
{
LPDRAWITEMSTRUCT lpDIS = (LPDRAWITEMSTRUCT)lParam;
SetDCBrushColor(lpDIS->hDC, RGB(255, 192, 203));
SelectObject(lpDIS->hDC, GetStockObject(DC_BRUSH));
RECT rect = { 0 };
rect.left = lpDIS->rcItem.left;
rect.right = lpDIS->rcItem.right;
rect.top = lpDIS->rcItem.top;
rect.bottom = lpDIS->rcItem.bottom;
RoundRect(lpDIS->hDC, rect.left, rect.top, rect.right, rect.bottom, 50, 50);
// button text
DrawText(lpDIS->hDC, TEXT("MOVE"), -1, &rect, DT_SINGLELINE | DT_CENTER | DT_VCENTER);
return TRUE;
}
return FALSE;
}
/*
* Call the target class' draw method
*/
void Facet::informPaint() {
elp_path->paint();
}
/*
* Say there is more than a single push button,
* this guy picks out the correct button that got clicked
* and calls the corresponding apt function
*/
bool Facet::actionPerformed(HWND hWnd, WPARAM wParam, LPARAM lParam)
{
switch (LOWORD(wParam))
{
case 12321:
elp_path->moveElliptic();
return TRUE;
default:
return FALSE;
}
}
Facet::~Facet()
{
delete elp_path;
}
C++ Animation Code for Elliptical Path - Header File
#define aWIDTH 10
#define aHEIGHT 10
class EllipticalPath
{
public:
EllipticalPath(HWND, int, int);
virtual ~EllipticalPath();
void paint();
void moveElliptic();
protected:
HWND hWindow;
HDC hdc;
int window_width;
int window_height;
COLORREF ball_colour;
//ellipse coordinates
int h;
int k;
int a;
int b;
int x;
int y;
HPEN ball_pen;
HBRUSH ball_brush;
};
C++ Animation Code for Elliptical Path - Class File
#include "EllipticalPath.h"
#include <math.h>
EllipticalPath::EllipticalPath(HWND hWnd, int window_width, int window_height)
{
hWindow = hWnd; // save away window handle
this->window_width = window_width; // save away window width
this->window_height = window_height; // save away window width
ball_colour = RGB(255, 0, 0);
//ellipse coordinates
h = 250;
k = 275;
a = 150;
b = 100;
x = h - a;
y = k;
// Pen and brush for travelling ball
ball_pen = CreatePen(PS_SOLID, 1, RGB(0, 0, 0));
ball_brush = CreateSolidBrush(ball_colour);
hdc = GetDC(hWindow);
SelectObject(hdc, ball_pen); // select ball pen colour
SelectObject(hdc, ball_brush); // select ball brush colour
}
/*
* draws the ball/circle using the apt color
*/
void EllipticalPath::paint() {
// draw a dot
Ellipse(hdc, x, y, x + aWIDTH, y + aHEIGHT);
}
/*
* Repeatedly draws ball so as to simulate a continuous motion
*/
void EllipticalPath::moveElliptic() {
// condition for continuing motion
while (x <= h + a) {
y = (int)round(k - ((double)b / a) * sqrt(pow(a, 2) - pow((x - h), 2)));
paint();
y = (int)round(k + ((double)b / a) * sqrt(pow(a, 2) - pow((x - h), 2)));
paint();
x += 20;
// introduce a delay between renderings
Sleep(50);
}
}
EllipticalPath::~EllipticalPath()
{
DeleteObject(ball_pen);
DeleteObject(ball_brush);
ReleaseDC(hWindow, hdc);
}