I have three points in 2D and I want to draw a quadratic Bézier curve passing through them. How do I calculate the middle control point (x1 and y1 as in quadTo)? I know linear algebra from college but need some simple help on this.
How can I calculate the middle control point so that the curve passes through it as well?
Let P0, P1, P2 be the control points, and Pc be your fixed point you want the curve to pass through.
Then the Bezier curve is defined by
P(t) = P0*t^2 + P1*2*t*(1-t) + P2*(1-t)^2
...where t goes from zero to 1.
There are an infinite number of answers to your question, since it might pass through your point for any value of t... So just pick one, like t=0.5, and solve for P1:
Pc = P0*.25 + P1*2*.25 + P2*.25
P1 = (Pc - P0*.25 - P2*.25)/.5
= 2*Pc - P0/2 - P2/2
There the "P" values are (x,y) pairs, so just apply the equation once for x and once for y:
x1 = 2*xc - x0/2 - x2/2
y1 = 2*yc - y0/2 - y2/2
...where (xc,yc) is the point you want it to pass through, (x0,y0) is the start point, and (x2,y2) is the end point. This will give you a Bezier that passes through (xc,yc) at t=0.5.
I have used Nemos answer in my JavaFX apllication, but my goal was to draw the curve, so that the visual turning point of the curve always fits with the choosen fixed one (CP).
CP = ControlPoint
SP = StartPoint
EP = EndPoint
BP(t) = variable Point on BeziérCurve where t is between 0 and 1
To achieve this i made a variable t (not fix 0.5). If the choosen Point CP is no longer in the middle between SP and EP, you have to vary t up or down a little bit. As a first step you need to know if CP is closer to SP or EP: Let distanceSP be the distance between CP and SP and distanceEP the distance between CP and EP then i define ratio as:
ratio = (distanceSP - distanceEP) / (distanceSP + distanceEP);
Now we are going to use this to vary t up and down:
ratio = 0.5 - (1/3) * ratio;
note: This is still an approximation and 1/3 is choosen by try and error.
Here is my Java-Function: (Point2D is a class of JavaFX)
private Point2D adjustControlPoint(Point2D start, Point2D end, Point2D visualControlPoint) {
// CP = ControlPoint, SP = StartPoint, EP = EndPoint, BP(t) = variable Point on BeziérCurve where t is between 0 and 1
// BP(t) = SP*t^2 + CP*2*t*(1-t) + EP*(1-t)^2
// CP = (BP(t) - SP*t^2 - EP*(1-t)^2) / ( 2*t*(1-t) )
// but we are missing t the goal is to approximate t
double distanceStart = visualControlPoint.distance(start);
double distanceEnd = visualControlPoint.distance(end);
double ratio = (distanceStart - distanceEnd) / (distanceStart + distanceEnd);
// now approximate ratio to be t
ratio = 0.5 - (1.0 / 3) * ratio;
double ratioInv = 1 - ratio;
Point2D term2 = start.multiply( ratio * ratio );
Point2D term3 = end.multiply( ratioInv * ratioInv );
double term4 = 2 * ratio * ratioInv;
return visualControlPoint.subtract(term2).subtract(term3).multiply( 1 / term4);
}
I hope this helps.
If you don't want the exact middle point, rather you want any value for t (0 to 1), the equation is:
controlX = pointToPassThroughX/t - startX*t - endX*t;
controlY = pointToPassThroughY/t - startY*t - endY*t;
Of course, this will also work for the mid point, just set t to be 0.5. Simple! :-)
Let the quad bezier we want to take as P(t) = P1t^2 + PC2t(1-t) + P2*(1-t)^2 and that quad bezier passing throw P1,Pt,P2
The best quad bezier that pass through the three points P1,Pt,P3 have the control point PC with tension directed in the perpendicular of the tangent of the curve. That point is also bisector of that bezier. Any bezier starting P1 and ending P3 with PC on the rect line that pass throw PC and Pt cuts the quad beziers at the same parametric t value.
The PC point is not achieved through the parametric position t=.5 of the bezier. In general for any P1,Pt,P2 the we got a Pc as described on the next formula.
The resulting PC is also the near point of that bezier to Pt and the rect line that pass through Pt and Pc is the bisector of the triangle P1, Pt, Pc.
Here is the paper where the teorem and formula is described -That is at my website https://microbians.com/math/Gabriel_Suchowolski_Quadratic_bezier_through_three_points_and_the_-equivalent_quadratic_bezier_(theorem)-.pdf
And also here is code
(function() {
var canvas, ctx, point, style, drag = null, dPoint;
// define initial points
function Init() {
point = {
p1: { x:200, y:350 },
p2: { x:600, y:350 }
};
point.cp1 = { x: 500, y: 200 };
// default styles
style = {
curve: { width: 2, color: "#333" },
cpline: { width: 1, color: "#C00" },
curve1: { width: 1, color: "#2f94e2" },
curve2: { width: 1, color: "#2f94e2" },
point: { radius: 10, width: 2, color: "#2f94e2", fill: "rgba(200,200,200,0.5)", arc1: 0, arc2: 2 * Math.PI }
}
// line style defaults
ctx.lineCap = "round";
ctx.lineJoin = "round";
// event handlers
canvas.onmousedown = DragStart;
canvas.onmousemove = Dragging;
canvas.onmouseup = canvas.onmouseout = DragEnd;
DrawCanvas();
}
// draw canvas
function DrawCanvas() {
ctx.clearRect(0, 0, canvas.width, canvas.height);
// control lines
ctx.lineWidth = style.cpline.width;
ctx.strokeStyle = style.cpline.color;
ctx.beginPath();
ctx.moveTo(point.p1.x, point.p1.y);
ctx.lineTo(point.cp1.x, point.cp1.y);
ctx.lineTo(point.p2.x, point.p2.y);
ctx.stroke();
// curve
ctx.lineWidth = style.curve.width;
ctx.strokeStyle = style.curve.color;
ctx.beginPath();
ctx.moveTo(point.p1.x, point.p1.y);
through = !document.getElementById("cbThrough").checked;
if(through)
{
tmpx1 = point.p1.x-point.cp1.x;
tmpx2 = point.p2.x-point.cp1.x;
tmpy1 = point.p1.y-point.cp1.y;
tmpy2 = point.p2.y-point.cp1.y;
dist1 = Math.sqrt(tmpx1*tmpx1+tmpy1*tmpy1);
dist2 = Math.sqrt(tmpx2*tmpx2+tmpy2*tmpy2);
tmpx = point.cp1.x-Math.sqrt(dist1*dist2)*(tmpx1/dist1+tmpx2/dist2)/2;
tmpy = point.cp1.y-Math.sqrt(dist1*dist2)*(tmpy1/dist1+tmpy2/dist2)/2;
ctx.quadraticCurveTo(tmpx, tmpy, point.p2.x, point.p2.y);
}
else
{
ctx.quadraticCurveTo(point.cp1.x, point.cp1.y, point.p2.x, point.p2.y);
}
ctx.stroke();
//new, t range is [0, 1]
ctx.beginPath();
ctx.lineWidth = style.curve1.width;
ctx.strokeStyle = style.curve1.color;
ctx.moveTo(point.p1.x, point.p1.y);
// control points
for (var p in point) {
ctx.lineWidth = style.point.width;
ctx.strokeStyle = style.point.color;
ctx.fillStyle = style.point.fill;
ctx.beginPath();
ctx.arc(point[p].x, point[p].y, style.point.radius, style.point.arc1, style.point.arc2, true);
ctx.fill();
ctx.stroke();
}
}
// start dragging
function DragStart(e) {
e = MousePos(e);
var dx, dy;
for (var p in point) {
dx = point[p].x - e.x;
dy = point[p].y - e.y;
if ((dx * dx) + (dy * dy) < style.point.radius * style.point.radius) {
drag = p;
dPoint = e;
canvas.style.cursor = "move";
return;
}
}
}
// dragging
function Dragging(e) {
if (drag) {
e = MousePos(e);
point[drag].x += e.x - dPoint.x;
point[drag].y += e.y - dPoint.y;
dPoint = e;
DrawCanvas();
}
}
// end dragging
function DragEnd(e) {
drag = null;
canvas.style.cursor = "default";
DrawCanvas();
}
// event parser
function MousePos(event) {
event = (event ? event : window.event);
return {
x: event.pageX - canvas.offsetLeft,
y: event.pageY - canvas.offsetTop
}
}
// start
canvas = document.getElementById("canvas");
if (canvas.getContext) {
ctx = canvas.getContext("2d");
Init();
}
})();
html, body { background-color: #DDD;font-family: sans-serif; height: 100%; margin:0;
padding:10px;
}
canvas { display:block;}
#btnControl { font-size:1em; position: absolute; top: 10px; left: 10px; }
#btnSplit { font-size:1em; position: absolute; top: 35px; left: 10px; }
#text { position: absolute; top: 75px; left: 10px; }
a {
text-decoration: none;
font-weight:700;
color: #2f94e2;
}
#little { font-size:.7em; color:#a0a0a0; position: absolute; top: 775px; left: 10px; }<h1>Quadratic bezier throw 3 points</h1>
<div>
Also take a look the the math paper <a target="_blank" href="https://microbians.com/mathcode">Quadratic bezier through three points! →</a> <br/><br/>
Gabriel Suchowolski (<a href="https://microbians.com" target="_blank">microbians</a>), December, 2012
</div>
<div id="little">Thanks to 艾蔓草 xhhjin for the code (that I fork) implementing my math paper.</div>
<br/>
</div>
<input type="checkbox" id="cbThrough" name="through"/>Primitive quadratic Bezier (as control points)</input><br/><br/>
<canvas id="canvas" height="500" width="800" class="through" style="cursor: default; background-color: #FFF;"></canvas>Enjoy
