Bezier curve revisited - an implementation using Javafx...

 Cubic Bézier curves...

Four points P₀, P₁, P₂ and P₃ in the plane or in higher-dimensional space define a cubic Bézier curve. The curve starts at P₀ going toward P₁ and arrives at P₃ coming from the direction of P₂. Usually, it will not pass through P₁ or P₂; these points are only there to provide directional information. The distance between P₁ and P₂ determines "how far" and "how fast" the curve moves towards P₁ before turning towards P₂.

Writing B_Pi,Pj,Pk(t) for the quadratic Bézier curve defined by points P_i, P_j, and P_k, the cubic Bézier curve can be defined as an affine combination of two quadratic Bézier curves:

${\displaystyle \mathbf{B}(t)=(1-t){\mathbf{B}}_{{\mathbf{P}}_0,{\mathbf{P}}_1,{\mathbf{P}}_2}(t)+t{\mathbf{B}}_{{\mathbf{P}}_1,{\mathbf{P}}_2,{\mathbf{P}}_3}(t),0\le t\le 1.}$

The explicit form of the curve is:

${\displaystyle \mathbf{B}(t)=(1-t{)}^3{\mathbf{P}}_0+3(1-t{)}^{2}t{\mathbf{P}}_1+3(1-t)t^2{\mathbf{P}}_2+t^3{\mathbf{P}}_3,0\le t\le 1.}$

Any series of 4 distinct points can be converted to a cubic Bézier curve that goes through all 4 points in order. Given the starting and ending point of some cubic Bézier curve, and the points along the curve corresponding to t = 1/3 and t = 2/3, the control points for the original Bézier curve can be recovered.

Here's how it looks like - an experimentation...

And here's the source code of the experimentation (JavaFx application)...

import javafx.application.Application;
import javafx.scene.Scene;
import javafx.scene.layout.Pane;
import javafx.scene.paint.Color;
import javafx.scene.shape.Circle;
import javafx.scene.shape.CubicCurve;
import javafx.scene.shape.Line;
import javafx.stage.Stage;

public class BezierCurveDemo extends Application {

    @Override
    public void start(Stage primaryStage) {
        // Main pane
        Pane pane = new Pane();

        // Control points
        Circle start = createControlPoint(100, 300, Color.RED);  // Start point
        Circle control1 = createControlPoint(200, 100, Color.BLUE); // First control point
        Circle control2 = createControlPoint(400, 100, Color.BLUE); // Second control point
        Circle end = createControlPoint(500, 300, Color.RED);    // End point

        // Lines to show the control points
        Line line1 = new Line();
        Line line2 = new Line();

        line1.startXProperty().bind(start.centerXProperty());
        line1.startYProperty().bind(start.centerYProperty());
        line1.endXProperty().bind(control1.centerXProperty());
        line1.endYProperty().bind(control1.centerYProperty());
        line1.setStroke(Color.LIGHTGRAY);

        line2.startXProperty().bind(control2.centerXProperty());
        line2.startYProperty().bind(control2.centerYProperty());
        line2.endXProperty().bind(end.centerXProperty());
        line2.endYProperty().bind(end.centerYProperty());
        line2.setStroke(Color.LIGHTGRAY);

        // Bezier Curve
        CubicCurve bezierCurve = new CubicCurve();
        bezierCurve.setStroke(Color.BLACK);
        bezierCurve.setFill(null);
        bezierCurve.setStrokeWidth(2);

        bezierCurve.startXProperty().bind(start.centerXProperty());
        bezierCurve.startYProperty().bind(start.centerYProperty());
        bezierCurve.controlX1Property().bind(control1.centerXProperty());
        bezierCurve.controlY1Property().bind(control1.centerYProperty());
        bezierCurve.controlX2Property().bind(control2.centerXProperty());
        bezierCurve.controlY2Property().bind(control2.centerYProperty());
        bezierCurve.endXProperty().bind(end.centerXProperty());
        bezierCurve.endYProperty().bind(end.centerYProperty());

        // Add everything to the pane
        pane.getChildren().addAll(line1, line2, bezierCurve, start, control1, control2, end);

        // Create scene and display
        Scene scene = new Scene(pane, 600, 400);
        primaryStage.setTitle("Bezier Curve Demo");
        primaryStage.setScene(scene);
        primaryStage.show();
    }

    // Helper function to create control points
    private Circle createControlPoint(double x, double y, Color color) {
        Circle circle = new Circle(8, color);
        circle.setCenterX(x);
        circle.setCenterY(y);

        // Make control points draggable
        circle.setOnMouseDragged(event -> {
            circle.setCenterX(event.getX());
            circle.setCenterY(event.getY());
        });

        return circle;
    }

    public static void main(String[] args) {
        launch(args);
    }
}