Demo code for calculating the touch point between a "ballooning" circle and a rectangle

BalloonTouch.cs

using UnityEngine;

[ExecuteAlways]
public class BalloonTouch : MonoBehaviour {

	public Transform start;
	public Transform other;

	void Update() {
		Vector2 startPoint = start.position;
		Vector2 dir = start.up;

		// Animate rect for demonstration
		other.position = new Vector3(2f * Mathf.Sin(Time.time * 2.3f), 2.5f + Mathf.Cos(Time.time));
		other.rotation = Quaternion.Euler(0f, 0f, Time.time * 90f);

		// Calculate score for rect
		float score = BalloonScoreRect(startPoint, dir, other, out Vector2 touch);

		// Draw debug visualizations
		float radius = 0.5f / score;
		DrawCircle(startPoint + dir * radius, radius, 48, Color.cyan);
		DrawCircle(touch, 0.05f, 12, Color.cyan);
		DrawCircle(startPoint, 0.05f, 12, Color.green);
	}

	static float BalloonScorePoint(Vector2 start, Vector2 dir, Vector2 point) {
		Vector2 vec = point - start;
		return Vector2.Dot(dir, vec) / vec.sqrMagnitude;
	}

	static float BalloonScoreLineSegment(Vector2 start, Vector2 dir, Vector2 p1, Vector2 p2, out Vector2 touch) {
		Vector2 lineVec = p2 - p1;
		Vector2 normal = new Vector2(-lineVec.y, lineVec.x).normalized;

		// This is what makes the ballon touch logic work.
		// By using a direction which is halfway between the input direction
		// and the (reverse) normal of the line, we get an intersection
		// which is exactly where the ballon would first touch the line.
		Vector2 intersectDir = (dir + normal);

		touch = Intersect(start, start + intersectDir, p1, p2);

		// Constrain touch point to be inside line segment.
		if (Vector2.Dot(touch - p2, p1 - p2) < 0)
			touch = p2;
		if (Vector2.Dot(touch - p1, p2 - p1) < 0)
			touch = p1;

		return BalloonScorePoint(start, dir, touch);
	}

	static float BalloonScoreRect(Vector2 start, Vector2 dir, Transform rect, out Vector2 touch) {
		void UseIfBetter(Vector2 p1, Vector2 p2, ref float score, ref Vector2 touch) {
			float newScore = BalloonScoreLineSegment(start, dir, p1, p2, out Vector2 newTouch);
			if (newScore > score) {
				score = newScore;
				touch = newTouch;
			}
			Debug.DrawLine(p1, p2);
		}

		Vector2 PA = rect.TransformPoint(new Vector2(1.0f, 1.0f));
		Vector2 PB = rect.TransformPoint(new Vector2(-1.0f, 1.0f));
		Vector2 PC = rect.TransformPoint(new Vector2(-1.0f, -1.0f));
		Vector2 PD = rect.TransformPoint(new Vector2(1.0f, -1.0f));

		float score = float.NegativeInfinity;
		touch = Vector2.zero;
		UseIfBetter(PA, PB, ref score, ref touch);
		UseIfBetter(PB, PC, ref score, ref touch);
		UseIfBetter(PC, PD, ref score, ref touch);
		UseIfBetter(PD, PA, ref score, ref touch);
		return score;
	}

	void DrawCircle(Vector2 center, float radius, int segments, Color color) {
		Vector2 p1 = center + new Vector2(radius, 0f);
		for (int i = 0; i < segments; i++) {
			int j = i + 1;
			float rad = j * 2f * Mathf.PI / segments;
			Vector2 p2 = new Vector2(Mathf.Cos(rad), Mathf.Sin(rad)) * radius + center;
			Debug.DrawLine(p1, p2, color);
			p1 = p2;
		}
	}

	static Vector2 Intersect(Vector2 line1A, Vector2 line1B, Vector2 line2A, Vector2 line2B) {
		// Line 1
		float A1 = line1B.y - line1A.y;
		float B1 = line1A.x - line1B.x;
		float C1 = A1 * line1A.x + B1 * line1A.y;

		// Line 2
		float A2 = line2B.y - line2A.y;
		float B2 = line2A.x - line2B.x;
		float C2 = A2 * line2A.x + B2 * line2A.y;

		float det = A1 * B2 - A2 * B1;
		if (det == 0) {
			// Parallel lines
			return Vector2.zero;
		}
		else {
			float x = (B2 * C1 - B1 * C2) / det;
			float y = (A1 * C2 - A2 * C1) / det;
			return new Vector2(x, y);
		}
	}
}

This demonstration is in the context of discussing automatic UI control navigation behavior based on a balloon approach. See the full discussion here:
godotengine/godot#103895

And my reply where I show a video of the code above here:
godotengine/godot#103895 (comment)