Develop a Flutter application with the objective of enabling users to manipulate the positions of five draggable points labeled as 1, 2, 3, 4, and 5 on the screen. The application should generate a smooth curve connecting these five points seamlessly. Additionally, the application should feature a ball with a diameter of 20 pixels on this newly generated screen. The task involves programming the application to extrapolate the movement of this ball along the drawn curve.
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Set Up Flutter Project:
- Create a new Flutter project.
- Set up necessary dependencies for drag-and-reposition, handling gestures on custom painters using touchable.
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Implement Drag-and-Reposition Functionality:
- Create draggable custom paint objects for each point (1, 2, 3, 4, and 5).
- Implement logic to handle dragging and repositioning of these widgets.
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Draw Smooth Curve:
- Utilize cubic Bézier splines to create a smooth curve that connects the five points.
- Implement algorithms or libraries that calculate the control points for the cubic Bézier curves based on the positions of the draggable points.
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Extrapolate Proof Ball Movement:
- Calculate the points on the curve using the PathMetric of the path.
- Implement logic to extrapolate the movement of the proof ball along the drawn curve with a Tween Animation from 0 to 1, representing the progress of the ball along the drawn curve.
- Track the positions of the proof ball for fitting them on the curve.
- Develop logic to calculate non-overlapping positions for the balls along the drawn curve.
The mathematical insights related to the cubic Bézier spline function, its derivatives, and boundary conditions have been provided in a PDF document. You can access it here.