To check whether two shapes interact with each other we use the ILcd2DBinaryTopology interface, as illustrated in the example below:

import com.luciad.geodesy.TLcdEllipsoid;
import com.luciad.geometry.topology.ILcd2DBinaryTopology;
import com.luciad.geometry.topology.ellipsoidal.TLcdEllipsoidalBasicBinaryTopology;
import com.luciad.shape.shape2D.ILcd2DEditablePoint;
import com.luciad.shape.shape2D.TLcd2DEditablePointList;
import com.luciad.shape.shape2D.TLcdLonLatPoint;
import com.luciad.shape.shape2D.TLcdLonLatPolygon;

public class BasicBinaryTopologyTutorial {

  public static void main(String[] args) {
    // Create the ellipsoid.
    // For this sample, it is hardcoded here.
    // In practice, the ellipsoid used by all shapes in the program should be used.
    // In a cartesian topology, no ellipsoid is needed.
    TLcdEllipsoid ellipsoid = new TLcdEllipsoid();

    // Create two shapes.
    // For this sample two simple triangles are created.
    // In practice, these ILcdShape instances could originate from any source.
    TLcdLonLatPolygon polygon1 = new TLcdLonLatPolygon(new TLcd2DEditablePointList(
        new ILcd2DEditablePoint[]{
            new TLcdLonLatPoint(0.0, 0.0),
            new TLcdLonLatPoint(10.0, 20.0),
            new TLcdLonLatPoint(20.0, 10.0)
        }, false), ellipsoid);
    TLcdLonLatPolygon polygon2 = new TLcdLonLatPolygon(new TLcd2DEditablePointList(
        new ILcd2DEditablePoint[]{
            new TLcdLonLatPoint(11.0, 12.0),
            new TLcdLonLatPoint(5.0, 4.0),
            new TLcdLonLatPoint(12.0, 0.0)
        }, false), ellipsoid);

    // Create the basic binary topology.
    ILcd2DBinaryTopology topology = new TLcdEllipsoidalBasicBinaryTopology(ellipsoid);

    // Test if the shapes interact.
    boolean result = topology.checkTopology(polygon1, polygon2, ILcd2DBinaryTopology.INTERACT);
    System.out.println("The shapes intersect: " + result);

The ILcd2DBinaryTopology interface only provides basic operations. See the How to check perform advanced (binary) topology checks tutorial for more advanced checks.