Demonstrates use of a JavaFX PerspectiveTransform - toggling or morphing the transform on and off.

PerspectiveMovement.java

import javafx.animation.*;
import javafx.application.*;
import javafx.beans.value.*;
import javafx.geometry.Pos;
import javafx.scene.*;
import javafx.scene.control.ToggleButton;
import javafx.scene.effect.PerspectiveTransform;
import javafx.scene.image.*;
import javafx.scene.layout.VBox;
import javafx.scene.paint.Color;
import javafx.scene.shape.Rectangle;
import javafx.scene.text.*;
import javafx.stage.Stage;
import javafx.util.Duration;

public class PerspectiveMovement extends Application {
  // perspective transformed group width and height.
  private final int W = 280;
  private final int H = 96;

  // upper right and lower right co-ordinates of perspective transformed group.
  private final int URY = 35;
  private final int LRY = 65;

  @Override public void start(Stage stage) {
    final PerspectiveTransform perspectiveTransform = createPerspectiveTransform();

    final Group group = new Group();
    group.setCache(true);
    setContent(group);

    final ToggleButton perspectiveToggle = createToggle(
      group, 
      perspectiveTransform
    );

    VBox layout = new VBox(10);
    layout.setAlignment(Pos.CENTER);
    layout.getChildren().setAll(
      perspectiveToggle,
      createMorph(perspectiveToggle, group, perspectiveTransform),
      group
    );
    layout.setStyle("-fx-padding: 10px; -fx-background-color: rgb(17, 20, 25);");

    stage.setScene(new Scene(layout));
    stage.show();
  }

  private void setContent(Group group) {
    Rectangle rect = new Rectangle(0, 5, W, 80);
    rect.setFill(Color.web("0x3b596d"));

    Text text = new Text();
    text.setX(4.0);
    text.setY(60.0);
    text.setText("A long time ago");
    text.setFill(Color.ALICEBLUE);
    text.setFont(Font.font(null, FontWeight.BOLD, 36));

    Image image = new Image(
      "http://icons.iconarchive.com/icons/danrabbit/elementary/96/Star-icon.png"
    );
    ImageView imageView = new ImageView(image);
    imageView.setX(50);

    group.getChildren().addAll(rect, imageView, text);
  }

  private PerspectiveTransform createPerspectiveTransform() {
    PerspectiveTransform perspectiveTransform = new PerspectiveTransform();

    perspectiveTransform.setUlx(0.0);
    perspectiveTransform.setUly(0.0);
    perspectiveTransform.setUrx(W);
    perspectiveTransform.setUry(URY);
    perspectiveTransform.setLrx(W);
    perspectiveTransform.setLry(LRY);
    perspectiveTransform.setLlx(0.0);
    perspectiveTransform.setLly(H);

    return perspectiveTransform;
  }

  private ToggleButton createToggle(final Group group, final PerspectiveTransform perspectiveTransform) {
    final ToggleButton toggle = new ToggleButton("Toggle Perspective");
    toggle.selectedProperty().addListener(new ChangeListener<Boolean>() {
      @Override public void changed(ObservableValue<? extends Boolean> observable, Boolean wasSelected, Boolean selected) {
        if (selected) {
          perspectiveTransform.setUry(URY);
          perspectiveTransform.setLry(LRY);
          group.setEffect(perspectiveTransform);
        } else {
          group.setEffect(null);
        }
      }
    });

    return toggle;
  }

  private ToggleButton createMorph(final ToggleButton perspectiveToggle, final Group group, final PerspectiveTransform perspectiveTransform) {
    final Timeline distorter = new Timeline(
      new KeyFrame(
        Duration.seconds(0), 
        new KeyValue(perspectiveTransform.uryProperty(), 0,  Interpolator.LINEAR),
        new KeyValue(perspectiveTransform.lryProperty(), H, Interpolator.LINEAR)
      ),
      new KeyFrame(
        Duration.seconds(3), 
        new KeyValue(perspectiveTransform.uryProperty(), URY, Interpolator.LINEAR),
        new KeyValue(perspectiveTransform.lryProperty(), LRY, Interpolator.LINEAR)
      )
    );

    final ToggleButton morphToggle = new ToggleButton("Morph Perspective");
    morphToggle.selectedProperty().addListener(new ChangeListener<Boolean>() {
      @Override public void changed(ObservableValue<? extends Boolean> observable, Boolean wasSelected, Boolean selected) {
        if (!perspectiveToggle.isSelected()) {
          perspectiveToggle.fire();
        }
        if (selected) {
          distorter.setRate(1);
          distorter.play();
        } else {
          distorter.setRate(-1);
          distorter.play();
        }
      }
    });

    return morphToggle;
  }
}

Answer to StackOverflow question: Stretching Polygon to other Polygon with Java